From the Parisian Mahayana Seminary lesson book, Year of the Buddha 2332:
'In the Gandhari original [gospel letters from the Buddhist kingdom of India] Antiochos is referred to as "Amtiyoko nama Yona-raja" (lit. "The Greek king by the name of Antiokos"), beyond whom live the four other kings: "param ca tena Atiyokena cature 4 rajani Turamaye nama Amtikini nama Maka nama Alikasudaro nama" (lit. "And beyond Antiochus, four kings by the name of Ptolemy, the name of Antigonos, the name of Magas, the name Alexander" Alright alright, I'll give it away. This is real. Except for the attribution, this was in fact copied from Wikipedia (today, Year of the Buddha 2556.)
"It is not clear in Hellenic records whether these emissaries were actually received, or had any influence on the Hellenic world. Some scholars, however, point to the presence of Buddhist communities in the Hellenistic world from that time, in particular in Alexandria (mentioned by Clement of Alexandria). The pre-Christian monastic order of the Therapeutae may have drawn inspiration for its ascetic lifestyle from contact with Buddhist monasticism, although the foundation and Scriptures were Jewish. Buddhist gravestones from the Ptolemaic period have also been found in Alexandria, decorated with depictions of the Wheel of the Law. Commenting on the presence of Buddhists in Alexandria, some scholars have even pointed out that "It was later in this very place that some of the most active centers of Christianity were established"'.
It's a bit odd that a Semitic religion ended up dominating Europe, and a blue-eyed Indo-European's religion ended up dominating East Asia - although oddly, not the land of his birth south of the Himalayas). But in the third century B.C., the Indian Buddhist King Asoka tried. After his conversion, he improved trade routes and sent missionaries throughout South Asia and the ancient Near East. In this he was like a Buddhist Constantine and Paul rolled into one; imagine a Buddhist New Testament with books named after letters to the evangelized city-states, like Alexandrians and Bactrians and Persians (instead of Romans and Galatians and Ephesians). The bottom image is an evangelical Buddhist inscription in Greek and Aramaic - by Asoka, from Kandahar, Afghanistan. Contact between Buddhists and the classical Near East always seem like a bit of alternate history to us modern Westerners.
Of course some of these monuments and markers have been destroyed by our throwback friends the Taliban, but they're just doing what good fundamentalists of all stripes do: think of the Spanish missionaries destroying Mayan texts, or early European Christians censoring and smearing classical materialist works, or any number of political book-burnings in the twentieth century. This brings up an obvious question: if Buddhism had its champion in a Asoka, then where are the Buddhist temples in Athens and Afghanistan today? The answer is obvious in retrospect when you consider religion as just another set of customs. If your philosophy (whether it appeals to the supernatural or not) is not traveling along at the head of a conquering army, or the merchants and diplomats of a powerful empire, the odds are against you if you don't have another trick, like getting endorsements from people in positions of power. (Scientology had the smart idea of spreading into people who have both influence, and weak intellectual immune systems.) It also helps for your philosophy to be intolerant of syncretism and pluralism, and here Asoka was too nice. He felt bad for having prosecuted a bloody war prior to his conversion, and while he did favor Buddhism, he did not punish non-Buddhists. Buddhism eventually did reach the rest of Asia - southeast Asia in Asoka's lifetime, and then China a few centuries later - by "organic" diffusion along the silk road or from missionaries sent out by the religion itself.
Again the differing history of religion in the Far East and the Middle East/Europe is interesting. It might not be anything about the pre-existing culture or geography or political systems of the regions, but rather the coincidental content of the religions themselves. Two innovations that the three Abrahamic religions happened to produce were 1) actively excluding other belief systems and 2) early in their history, successfully infiltrating existing secular powers. Indeed the Abrahamic religions got progressively better at this as time went on. The Jews kept mostly to themselves except during military occupation, then the Christians grew to dominate Rome after a few centuries, and finally Mohammed seems to have conceived Islam as a means to political and military power right from the start. Islam - Abrahamic religion v3.0 - was the best one so far. It's also probably no coincidence that it's the cultural and geographic crossroads of the Middle East where these innovations appeared. A religion that isn't a strong competitor right out of the cradle isn't going to get very far in a place like that!
So there was no Gupta army storming west out of India to force Buddhism onto the Persians and Greeks and Romans, partly because Buddhists are not required to exclude other beliefs. Fair enough; and incidentally, some of the Mongol armies were Buddhist, and some followed an indigenous Mongolian religion, but again, neither of these required conversion. If you paid your taxes the Mongols didn't care. That's why Russians today don't follow the sky god Tengri. (The euphemism "indigenous religion" just means "a religion that's not one of the few indigenous religions that escaped the ethnic group that created them and then spread globally".) But this leaves unanswered the opposite question, which is why India and China aren't Christian or Muslim today. If Alexander had crossed the Indus - or a Chinese-Turkic empire had controlled the Middle East - then very likely whatever religions appeared in this region would have spread east at least as much as they spread west. (In a stable, united post-Alexandrian Eurasia, my money is on a prophet appearing and spreading his faith a little earlier in history.) But we should also remember that we're still in medias res of the global diffusion of ideas, and it's possible that the monotheistic, active-excluding religions just haven't had enough time to crowd out the tolerant ones with tolerant leaders. That is to say, the world's gardens haven't yet all been colonized with the most hardy invasives on offer. Of course, the parts of Asia that came into contact with Abrahamism v3.0 are, in fact, Muslim today.
In closing, modern Korea is a much more interesting case. One half of it has its own brand of exclusive Korean-nationalist communism - originally a European philosophy; how syncretic - which tolerates no (other) religion - and the other half appears very much like it's in the process of becoming Christian, complete with politically ascendant creationists trying to impose restrictions on what is taught in biology classes. And all of this in less than a century.