Laïcité is basically the French form of secularism. It is more ''aggressive'' than the typical secularism found in Anglo-speaking countries. The state takes an active approach to keep religion out of the ''public sphere'' and religion is seen something to be kept in the ''private sphere''. For example, in government buildings, people can't display overt religious symbols which also includes public schools. Right-wing critics of this claim the French government of making religion ''politically incorrect''.
I know the French attuide towards religion is very different from the American one, you'll have to change lot of things for this occur. In France, due to the French Revolution, where the modern ''French identity'' was created, religion was seen as a obstable to attaining more freedom and equal right. Religion had a certain reputation of almost being synonymous with feudalism among some circles.
What changes would America need to have such a stance?
I think the relevant differences between America and France in this respect are:
(1) Many of the early American settlers came precisely to get away from government-enforced interference in their religious practice (in the form of compulsory membership of the Church of England and the like), so they and their ideological descendants would be likely to see a French-style laïcité less as "A welcome protection from the tyranny of the Church" and more as "Precisely the sort of interfering crap we came over here to escape from".
(2) France was overwhelmingly Catholic, whereas America was split into a multitude of different denominations. This means that, even if one particular Church (the Anglicans, say) ended up in an analogous position to the Catholic Church in France, it would be implausible to take this as a problem with "religion" or "Christianity" in general, and any resentment would probably be focused on the Anglican Church specifically.
(3) Gallicanism meant that the Catholic Church in France was much more influenced by the secular government, so resentment at the latter easily boiled over into resentment of the former. America never had a similar tradition of state control over the church, so there was no reason for resentment at government policies to lead to resentment of the Church.
(4) The prospect of a reactionary counter-revolution seeking to undo the French Revolution was seen as a real risk, and since the Catholic Church for a variety of obvious reasons wasn't very keen on the FR, they were seen by a lot of liberals as potential fifth columnists seeking to bring back the bad old days. Conversely, there has never been any notable body of opinion in America for undoing the American Revolution, so the idea that "If we let the Church do what it wants, it'll just try and bring back the King of England to rule over us" would come across as pretty laughable.
So, to get America to adopt a French model of laïcité, you'd have to make America more like France in these respects. A good start would be to make the Thirteen Colonies overwhelmingly Anglican somehow, so that in the minds of most people "religion" and "the Church of England" are more-or-less interchangeable. Then give the secular authorities control over the Church in America somehow -- give governors the authority to appoint bishops, that sort of thing. Then give the American Church pro-British sympathies after the Revolution, so that the idea of the Church supporting a return to British rule is a plausible one. Then you'd probably get something like laïcité introduced in America.