The main problem with armor isn't a lack of ability to stop bullets; continued use of armor by cuirassiers and the occasional infantryman through WWI shows that, yes, you can stop a bullet if you put enough steel between yourself and the gun.
The issue that forced out armor is the same as the issue that forced in guns, though: economics. Gothic plate was expensive. Even just breastplates or mail also start getting very, very pricey when you're equipping tens or hundreds of thousands of men with them.
Armor is also heavy. In the early modern era, soldiers were increasingly expected to carry their own supplies, and while a hoplite could run a marathon in 25kg of bronze, I doubt that the average peasant could march 30 km a day for a month while carrying spare socks, ammunition, molds, powder, food...and whatever armor his commander's decided to inflict on him. Note that armor in the post-Renaissance age was mostly used by cavalry, or by people in static positions (like the limited deployment of armor in WWI). In WWII, flak jackets were used primarily by airmen and naval crew, and not infantry because they were so heavy.
The fact that body armor started to get reintroduced on a wide scale in the 50s, after the introduction of plastics, aluminum and advanced ceramins, and the reduction of military sizes (relatively), strengthens my points, I think.
So, in order to get armor to stay widespread, you'd need to find something cheap and light to make it out of. Maybe somehow make the silk industry huge? I think that high-grade ceramics are impossible without 20th century chemistry, but maybe someone can make a good PoD.