In American English, and most other English dialects, "public school" means "school run by the government". However, in British English (and also I believe Indian, maybe Irish too), "public school" has a rather opposite meaning – "expensive private school for the children of rich". In no major dialect of English does "public school" mean "the broad modern standard of putting every child in school"
What you are really talking about is compulsory education – laws that oblige parents to send their children to a school, whether government-run or private.
Are Protestants responsible for compulsory education? It is true, that historically speaking, Luther and other Reformers did play an important role in promoting the idea in the early modern period - but they didn't invent it. It existed in ancient Sparta, for boys. Plato, in his Republic, advocated compulsory state-run education – not just for boys, but for girls too.
It is true the first few states in modern Europe were to enact it were Protestant, but Catholic Europe soon copied the idea – Protestant Prussia introduced it in 1763, but Catholic Austria followed suit only 11 years later (1774), while it took Protestant Sweden another 68 years (1842). The Ottoman Empire made education compulsory in Istanbul in 1824 (they planned to gradually roll it out empire-wide, but that took far longer to achieve than they'd initially anticipated); at that point, no US state had yet done so, the first being Massachusetts in 1852. (Although publicly funded education was established in Massachusetts over two centuries earlier, it wasn't compulsory, there was no legal obligation on parents to send their children to school until then.)
So, while certainly Protestants played an important role in promoting the idea at in the modern period, they didn't invent it, and some Catholic and Muslim jurisdictions introduced it before many Protestant jurisdictions had.