Without the news of the new world, there would be no more expeditions across the Atlantic Ocean for a very long time. The fact that Columbus did not return from his ill-fated and poorly conceived voyage was proof, in the minds of skeptical Europeans, that going West across the Atlantic was foolish.
Portugal was working on developing the route to India by going around Africa, and began to establish colonies along the way. Spain would have no choice but to join their efforts. Portugal would take the lead in all explorations and merchant activities, and eventually come to dominate Spain. Portuguese princes would occupy the throne of Spain for many years. Eventually, the Iberian Peninsula would fall to the expansion of the Ottoman Empire, and return once more to Islamic control.
Portugal's African colonies, ironically, would last longer than Portugal itself. However, over the years each would fall in various ways. Cut off from European support, their only choices were to integrate into the local politics and culture, or be conquered by the expanding African nations, grown rich and powerful through trade with the Ottomans. The Turks would only be able to control Northern Africa, blocked by the vast Sahara, but trade and Islamic missionaries would follow the Nile into the continent's heartland. Alliances and treaties would be made, and both sides would prosper.
The heir to the throne of Spain after the death of Isabella, her daughter Joanna the Mad (in Spanish Juana la Loca) would never marry Philip the Handsome, of the powerful House of Habsburg of Austria. The Habsburgs were only interested in power and wealth, and without the discovery of the Americas, Spain had neither. Manuel I of Portugal would likely have married Joanna rather than her sister, Maria of Aragon, under these circumstances. Thus, Charles V would never be born, and never become Holy Roman Emperor and one of the most influential rulers of the early Renaissance.
Indeed, the Renaissance itself would not have happened without the discoveries and wealth from the New World, leaving Europe mired in a prolonged intellectual darkness. Only much later, having conquered continental Europe, would the Ottomans invest in the progressive enlightenment in the arts and sciences that we would consider to be an avatar of the Renaissance.
Without the wealth of Aztec gold and Inca silver to buy cannons and mercenaries, the Habsburgs would eventually lose their long-running fight against the Ottoman Turks. Suleiman the Magnificent laid siege to Vienna in 1529 and in this universe conquered it.
Having taken Austria, the Turks would move on to take Venice and the feuding states of Northern Italy. From there it would be relatively simple to march on Rome. The Pope would flee to France, under the protection of King Francis I, along with Michelangelo and any others deemed important.
Since the devoted Catholics of the Spanish crown would not be sending lavish tithes to the church, the indulgences of the Papacy would not provoke Martin Luther to start the Protestant Reformation. Thus, even in exile in France, Pope Clement VII would still hold considerable influence over Christian Europe. Being a member of the Medici family also gave him secular resources as well.
King Henry VIII of England would have the advantage in this situation. With France under siege by the Ottomans, the military might of England would make him a desperately needed ally. Always the pragmatic politician, Pope Clement would have eagerly granted Henry the divorce he craved, in exchange for an alliance against the Turks. However, since Catherine of Aragon would not have the massive dowery from Spain that England so eagerly desired, it is unlikely that Henry would have married her in the first place. Also, given that the Boleyn sisters were part of Catherine's entourage when Henry met them, he would probably not have married Anne, either.
In any case, England could remain steadfast against the Turks, safe against attack as an island nation. The Papacy would relocate to London after the fall of Paris.
In the middle of this conflict were the German states of the Holy Roman Empire. The devastating winters of the Little Ice Age had made it rough for the farmers of Northern Europe since 1550. Without the potato, a hardy vegetable from the New World that could survive the cold weather far better than the wheat and barley of that region, famine would decimate Northern Europe. Most of the North would fall without a shot being fired. Turkish Pashas, obedient and loyal to the Sultan in Istanbul, would replace the Northern European kings for a promise of food in exchange for military aid.
The Little Ice Age continued until 1850. During that time, the Ottomans would work on improving their infrastructure, and gradually chip away at the last strongholds of the Christian world throughout continental Europe. The Ottomans built roads, so that food could be transported from the more fertile lands around the Mediterranean to the starving North. Manufacturing would replace farming along the Rhine valley, and German steel howitzers would replace the brass cannons of the Turkish army.
Under Ottoman rule, Europe could prosper and thrive. The cities would be kept clean and plague-free. Commerce with China and India would proceed unchecked, since the Turks controlled the Silk Road. When rebellions sprang up, as they always do from time to time, the local Janissaries could put them down quickly and ruthlessly. However, for the most part the Turks would rule with a light touch. Conversion to Islam would be encouraged, but Christians in the Ottoman Empire were typically allowed to co-exist in many regions, as long as they did so peacefully.
Many of the educated and (formerly) wealthy Jews that were driven out of Spain were rescued and welcomed by the Turks. Even in our own time-line, the Sultan's court physicians were all of Jewish faith for hundreds of years. In the World without Columbus, Jewish persecution across Europe would effectively end with the coming of the Ottoman Empire. The Turks had no sympathy for religious intolerance.
The writings of Copernicus, and later Galileo and Bruno, met with great favor among Turkish astronomers, who had been studying the heavens with more open-mindedness than their Christian counterparts for hundreds of years by that point. Indeed, Galileo Galilei would become a court astronomer, and serve under three different sultans before his death in 1642.
In 1742, just 150 years after the failed voyage of Columbus, a desperate Irish mission would be sent to find new lands for trade or colonization. They would come upon Newfoundland, and then the Northeastern coast of North America. Contact would likely be with the Athabaskan tribes at first, but the Kanonsionni (known to most of us today as Iroquois) would be the dominant force in the Northeastern forests, having grown and prospered without interference from the Europeans, eventually expanding their confederacy to more than a hundred nations. The nature of such expansion would have forced the Kanonsionni to refine and perfect their concept of a democratic, representative government, the like of which had never been seen before.
Not coming as conquerors, but explorers seeking trade and allies, the European contact would be peaceful. England and the Kanonsionni nations signed treaties, and trade blossomed between them. The sharing of wealth and knowledge, including technology, would profit on both sides.
The recent discoveries of vaccination and the process of pasteurization would check the spread of disease from the Europeans to the previously unexposed people of the New World. Therefore, later European immigrants to the New World would not find it empty and depopulated, but thriving with millions of native inhabitants. Some European ways would be adapted or even embraced eagerly – such as the creation and use of firearms, but the cultures, languages and religions of the New World peoples would remain relatively unscathed.
Writing is also eagerly adapted, but the Hundred Nations is also home to several hundred languages. After years of heated debate, the Kanonsionni elders decide that Cherokee, or Tsalagi Gawonihisdi in that language, or Tsalagi for short, will be their common official language. Printers and publishers quickly adopt the standard, and even European and Turkish books are translated. Within a generation, the Hundred Nations becomes the home of the most literate people on Earth.
In turn, the Kanonsionni model of a democratic republic spreads throughout Europe like wildfire. In 1799, the English House of Commons is established. Switzerland, the only other European nation to remain free of Turkish rule, became a direct democracy, and the Swiss Federal Assembly met for the first time in 1850. Other European nations under the thumb of the Ottoman Sultan began to speak of freedom and democratic rule. This did not go unnoticed.
-- From the Afterword of Burning Woman and the Ghost Lance, available on Amazon.com