I think it was more the other direction. Protestantism was based on the idea that people should read the Bible themselves rather than have priests interpret it for them as had been traditionally done. This was impractical before printing because hand-copied books were so expensive. Cheap printed books led to the possibility of these movements being successful.
It goes both directions. Without the printing press, the rise of mass religious movements prioritizing literacy would not have been possible. But once the printing press was there, the religious movements pushed literacy forward faster than it would have gone without them.
The Communists came pretty close to dumping characters in favor of Latin script (pinyin). And this actually happened in Vietnam, although we have the French to thank/blame for that.