It's only in recent years that he's been consistently portrayed as a white man by Western (particularly American) Christians.

I'm sorry, but this claim of /u/best_of_badgers is just not true. They've cherry-picked individual examples without any sense of their historical context and skipped an entire millennium of western Europe. A millennium in which, if David Nirenberg is correct, Christian icongraphy was a major vehicle for working out concepts of racial difference before "race."

Medieval European (and Middle Eastern, for that matter) scholars inherited from antiquity the idea that climate affects the development of those native to the region--this will play an important role in the evolution of scientific racism into the early 20th century. The closer to the equator, the hotter the temperature--the angier, earthier, and darker-skinned people got. Centuries before "black" and "white" accrued their modern American meanings, medieval Latin writers were describing and illuminators were painting white and black people. There are a few cases, like the famous Spanish manuscript known as the Book of Games, where the depictions seem to mirror the diversity of the painters' experience to some extent: the illuminations feature dark-skinned and light-skinned Saracen men, and light-skinned Christian and Saracen women alike (the Christian men are all light-skinned). But especially when depicting situations of Christians vs. Saracens, as in illustrations to the Song of Roland, the Saracens are uniformly dark-skinned and the Christians light-skinned. The association of Christians as white and Muslims as black persists in literary descriptions. In medieval romance, characters with one Christian and one Saracen parent are even described as black and white spotted. Or when a Saracen character converts to Christianity, their skin color physically changes from black to white.

So European artists were well aware that native residents of more southern lands tended to have darker skin. And yet western European medieval Jesus is white. It's 14th century mystic Birgitta of Sweden who gives us, through her visions, the tradition of Mary as not just white but blonde. Her Jesus, too, shines with white light.

I want to highlight a second iconographical development over the later Middle Ages. Iconography is all about the use of visual shorthand to convey meaning. In earlier art, into the high Middle Ages, depictions of Jews are recognizable above all by their hats, sometimes also by their clothing. But over the later M.A., Jews start to take on what we now see as "stereotypically" Jewish features (stereotypically thanks to Christian-incubated iconography), primarily darker features and a hooked nose.

Medieval Christians knew Jesus was Jewish. Some theologians and pastors occasionally made a big deal out of it (typically in an anti-Jewish context). Do medieval illuminations and woodcuts of Jesus and Mary give them the stereotypical hooked nose?

No, they bestow that feature on demons and the devil.

Depictions of Jesus are iconographical: they represent concrete theological and often political claims. In western medieval art, the depiction of Jesus as "like us" reigned supreme and served to demarcate, isolate, and condemn everyone else.

Follow-up (which is maybe also a partial answer): did early Christians depict Jesus at all? I've always assumed (but never actually verified) that they followed the traditional Jewish prohibition on such things.

See my post here: https://www.reddit.com/r/AskHistorians/comments/4t73k5/when_did_artists_first_start_portraying_jesus/d5fqpj9

The "white" Jesus is based on Roman heroes and deities, particularly Jupiter and Apollo. Artists used preexisting iconography; there was no effort to imagine him realistically until recent times.

There is another good thread about Jesus on the front page of this sub right now, in which it's explained that Jesus was barely even referenced in primary sources at all until more than 100 years after his death. Thus, I'm not sure what the first known rendition of him is, but it seems likely that it was not made by anyone who had actually seen Jesus alive.

The earliest that I'm aware of is the good shepherd, where he's indeed shown with darker skin. here is another, from the fourth century.