Legally, Congress cannot do that, as long as it is "respecting" some religion. It's true that the US Constitution is more used as an excuse than upheld, but at least the language is pretty plain.
On the one hand there is a clear and comforting consensus in the US that they aren't going to force religious mores on anyone. So I don't see how the Congress could or would try. But if we ever end up in a situation where the US Congress decides it wants to go straight Catholic I really don't see how it would fail. They've got far too much power and they'll eventually rotate in a compliant judiciary and then redefine what words mean so they can do it.
It is a bit difficult when dealing with such an out-there scenario, but if Congress decides the US military should do something wild then eventually the military is going to do it. The realistic protection is that by virtue of good system design there isn't a plausible scenario where Congress would want to impose Catholicism on the US population by force.
However, I do think there is an expectation in Americans' minds that Congress is governed by the Constitution, and believing makes it so. Mostly.
The impeachment process requires a majority of the House and two thirds of the Senate, and then any replacement would have to be nominated by the President rather than Congress. Meanwhile the existing decision stands, and even the dissenting Supreme Court Justices (if there were any) should follow it as a result of stare decisis, and might even be inclined to because removing their colleagues under those circumstances could make them uneasy.