I don't quite see how you are saying a lie was related, but yes, you are right about republics.
The federal government can pass laws in congress, there are certain things they have responsibility for while the states are responsible for other things.
In terms of the Supreme Court, it is a little different than the UK, they have more power to stand against Congress. But they can basically make judgments that reflect constitutional principles. Other things are up to the states or state courts.
It's closer to the EU and the countries within it or maybe something like the Scottish Parliament in the UK.
So a lot of the job of the SC is to determine whether laws are within what the constitution allows or not. In RvW, it was argued that the state abortion laws were not in accordance with the right to privacy as outlines in the Constitution. So all those laws were struck down and it restricted the scope of what laws states could make.
But when this judgement was reviewed in a new case, it was found to be unsound - the judges said the ruling was wrong, that the right to privacy does not apply in that way.
The reason for the change mainly has to do with there being two main schools of thought in terms of how loosely the constitution is interpreted. That being said, the original judgement was always been considered a pretty weak interpretation by constitutional scholars, which is to say it seemed forced and didn't fit in with other uses of the right of privacy.
If the Democrats wanted a federal law, they needed to have congress pass it. But that would have involved convincing people that it should be a federal rather than state level law.