To expand on LS’s great post, we’re bound by our Constitution here, as you probably know. It is up to our Supreme Court to determine whether or not a law is constitutional or not. In 1973, the case of Roe v. Wade came before the Supreme Court. The Court, in a 7-2 majority, upheld Roe v. Wade and declared abortion every US woman’s constitutional right. Abortion became legal in all 50 states.
The president cannot sign an order striking down Roe v. Wade. Congress cannot legislate against it. The people who want to stop abortion want very conservative judges appointed to the SC so that maybe, some day, these conservative judges will strike down Roe v. Wade and abortion will not then be legal in the US.
There seems little chance of that happening, though. Five of the justices who upheld Roe v. Wade were conservative, and strict constructionists as opposed to liberal or fluid constructionists. Five were Republican appointed justices. So, being conservatives, and probably strict constructionists, they had to be pretty convinced that abortion on demand did not violate the US constitution. Subsequent decisions, heard under a different mix of justices, have upheld Roe v. Wade. One of the most recent was 1992, Planned Parenthood v. Casey. Again, a Republican appointed court upheld Roe v. Wade.
So, it seems that those wanting abortion on demand made illegal in the US will have to take a different path for that to happen.