No, my pantomime is of me signing, complete with little flourish at the end. If I want someone to roll their car window down, I do the “roll down” pantomime, not the “poke a button” pantomime.
Now if I could just get out of this damn invisible cube.
These little pantomimes do eventually change though. These days the universal sign for a phone is to hold up a hand with a thumb to an ear and a finger pointing to the mouth. Fifty years ago, the common sign was winding an invisible handle, even though only a few could actually remember when you had to wind a telephone handle to contact the operator.
I suspect that in the future the bill will be displayed on your mobile communication device and paid by a tap of a finger and car windows will open or close on voice commands. There may be some very expensive restaurants where your food is cooked and served by real people, but they will be few and far between.
D’oh! That never occurred to me.
When I’m in the pub this evening, I think I’ll ask them for my invoice, just to see what happens.
Funny. I always make a check mark 
in the air. I’ve seen others do it.
Sorry to diverge a bit.
You don’t say check in a serious game of chess. (I said it once to a grandmaster and he told me off afterwards!)
Check simply means your opponent’s King is attacked. It has little to do with potentially winning moves.
The Persian for King is ‘Shah’. It’s a reasonable belief that ‘checkmate’ comes from the overheard Persian phrase ‘Shah Mat’, which means ‘The King is dead.’
(It’s also reasonable that the Rook in chess is named after the Persian word ‘Rukh’, meaning chariot.)
Or for a young American expat there who gets an entry level office job. “The American’s in the mailroom!”, people would quip.
In a related vein to the OP: Why do we use the word “bill” to mean a piece of paper currency? Elsewhere they are called “notes”. Holding a note is good, since it represents some other person or entity’s obligation to pay you or give you something in return. It is the opposite of a bill in much the same way as a check would be.
Come to think of it, an outstanding check you have just written is really more like a bill from your own perspective than it is like holding currency or a loan note.
Much less useful internationally, though; even where you share a spoken language. In the UK, for instance, that’s called a tick, which would require two levels of decoding to associate with you wanting the bill.
The only situation in which I can recall seeing that gesture here is when the person making it feels they’ve scored a rhetorical point against someone.
I suspect it’s related to the customer checking over the items on their bill, to make sure they were billed for the food they actually received. I can’t find a definitive cite to confirm this, though. But in roughly chronological order the word seems to have gone from “threatening (and thus restricting the movements of) the king in chess” to a general sense of “restraining something” (keeping it “in check”), to “preventing error” (“checking” a list), to “restaurant bill” – while still holding on to the older meanings as new ones were added. See check | Etymology, origin and meaning of check by etymonline
Edited to add: In other words, what UDS said back in post #4. 
nm
I prefer the phrase “tab me out” when I’d like to check bill…