No, the GFA mentions the European Convention on Human Rights (ECHR). But this was not intended to be the only authority since the GFA states the UK Government has complete sovereignty over Northern Ireland as well.
That is why the GFA states ".....and any Bill of Rights". Quite why you think any UK Bill of Rights would both contravene the GFA and/or that Northern Ireland wouldn't be included in it, I don't know. As per the GFA, Northern Ireland would get a section within it to itself which would take account of things there.
I guess this is like those who mistakenly thought that Brexit would breach the GFA until the High Court in Belfast ruled it didn't as the GFA was not actually dependent on continued EU membership. By the same token, the GFA is not dependent on continuing membership of the ECHR and the UK Government can bring in "any Bill of Rights for Northern Ireland" since it has sovereignty here.
So, far from the UK Government being on a path to criminalising abortion, its behaviour is very clearly the opposite as it has legalised abortion in Northern Ireland. And making moves to obligate for their services to be provided.
This is not a case of the US doing something and the UK following like a poodle. The UK has diverged from the United States in many areas over the years.