Is This Even A Logical Fallacy?

I’ve had many discussions with science deniers over the years.  One of their weirdest arguments is that gravity doesn’t exist.  

First, for the purposes of this discussion, let’s ignore the fact that gravity has always been an acceleration resulting in a force, and the obvious “if not gravity then what” question.  The latest incarnation goes like this:

- Newton’s Law of Universal Gravitation quantifies the acceleration due to gravity which results in a force.

- In General Relativity, Einstein shows that gravity causes the curvature of space-time thus proving gravity “is not a force” discrediting Newton’s Law of Universal Gravitation.

- Then discredit General Relativity.

- Conclusion, gravity doesn’t exist.

Summary:  Use General Relativity to discredit Newton.  Discredit GR with much hand-waving nonsense.  Claim gravity doesn’t exist.

Seriously, that’s the current argument.  The counter argument is fairly easy.  Of course, their typical response to the counter argument is “nuh uh”, because “reasons”.  

Aside from all of that, is this argument actually a logical fallacy?

The argument seems to be—General Relativity shows that Gravity is not a force in the Newtonian sense.  Therefore, gravity doesn’t exist.  

this is a Non Sequitur; the conclusion doesn’t follow from the premises.  It’s also a False Dilemma; it implies that there are only two possibilities—Gravity is either a Newtonian force or Gravity doesn’t exist. It excludes any other possibility 

This is fallacious as even from a purely conceptual standpoint, it makes no sense.

If A implies that B is false, if A is false, that doesn’t necessarily mean B is false as well.

If General Relativity is false, then it’s clause about Newtonian Gravity doesn’t mean anything as GR is false. They don’t affect each other unless one of them is true.