A search revealed a thread from about a year ago asking if this was common in the US; no one thought to ask why the term is used at all, a shameful lack of curiosity on our part. Does anybody know?
There doesn’t appear to be a single answer and the usages are all tangled together; e.g., “to check out” appears to derive from “getting the check.”
But I think it’s all based on some form of “checking the tab or total” at the end of a meal or a stay or other situation where charges are listed on an open tab for the duration.
WORD HISTORY:
The words check, chess, and shah are all related. Shah, as one might think, is a borrowing into English of the Persian title for the monarch of that country. The Persian word shh was also a term used in chess, a game played in Persia long before it was introduced to Europe. One said shh as a warning when the opponent’s king was under attack. The Persian word in this sense, after passing through Arabic, probably Old Spanish, and then Old French, came into Middle English as chek about seven hundred years ago. Chess itself comes from a plural form of the Old French word that gave us the word check. Checkmate, the next stage after check, goes back to the Arabic phrase shh mt, meaning “the king is dead.” Through a complex development having to do with senses that evolved from the notion of checking the king, check came to mean something used to ensure accuracy or authenticity. One such means was a counterfoil, a part of a check, for example, retained by the issuer as documentation of a transaction. Check first meant “counterfoil” and then came to mean anything, such as a bill or bank draft, with a counterfoilor eventually even without one.
Okay, that explains why it’s used as a substitute for ‘bill’. But why only in restaurants?
Why is the bill you run up in a bar the only place it’s called a tab? (Probably from “tabulate,” meaning essentially the same thing as “[to] check.”)
I think it’s just subtle evolution of the word usage, probably without a good or final explanation.
And why only in America?* Here in Britain we pay restaurant bills, the same as we pay electricity bills, gas bils, grocery bills, etc.
*Maybe it is not only in America. I do not know what Canadians, Australians, New Zealanders, etc. say they pay in a restaurant, but, it is plausible, because, in most cases, American English diverges further from British English than other versions of the language do, and most divergences in those other countries (Canada perhaps excepted) are in different directions from the American ones.
The word check used in restaurants is chiefly a U.S. invention of the 19th century.
Hmm. So you’re saying that in Toronto, when I say “cheque please” (which I do), I’m really saying “check please”?
That’s a real question.
Unless you’re playing chess. 
Personally, I never ask for the “check” in the restaurant; it’s always the “bill”. Don’t know how typical that is for Canadian English.
Sure, in Regina. But in the big city . . . 
Greater Toronto Area resident here. I always ask for the bill too.
I’m a US citizen and sometimes I ask for the bill, sometimes I ask for the check (when I have to ask, which is rare). Sometimes I just say, “we’re all finished eating now, and golly gee willigers we’re ready to go” which also gets the point across.
Which is perhaps a clue - “waiter’s check” is likely the device by which the waiter checked your order, probably in all aspects: its ordering, delivery and payment.
Restaurants are pretty much the only place I can think of where people commonly use sign lanuage to ask to settle their balance. The handy air waved “check mark” delivered with a flourish upon achieving eye contact with the waiter is a perfectly respectable gesture.
If we called it a “bill” we would be forced, we would have no other options, we would be forced to make eye contact with the waiter then contort a “duck face” (to signify “bill”). This is a wholely buffoonish expression to be making in polite company.
I am therefore thankful for use of the term “check”.
Here, in a part of the world where we call it a bill and not a check, we signal for the bill with a hand motion that imitates signing something with an invisible pen. Presumably it refers to signing the credit card slip, although in the era of Chip-and-PIN one doesn’t actually have to sign any more.
I have a feeling that the “hand flourish” for “check/bill, please” predates credit cards, and therefore must have developed first in the US, signifying the “check mark,” and wasn’t adopted in “bill” countries until after credit cards became common.
But this is GQ, so feelings don’t count! Anyone have the facts?
On language variety, I’m currently in Japan for the first time, and I’ve found that the term used here is チェック (chekku) for a restaurant bill. Doubtless that was borrowed from American English.
In the olden days, 19th century style, only the fairly well-to-do could afford to have dinners of any kind. Poor people ate at home or lived in boardinghouses that served meals. Therefore, anyone rich enough to eat out was rich enough to be trustworthy. And rich enough not to have to be bothered by pulling out money and counting it at the table or, worse, squabbling about who ordered what and splitting the bill, indicating a lack of wealth. The payee would be presented with a slip for the total of the meal, which was signed with a flourish with a gaudy fountain pen. Then at the end of the month the restaurant would send over a bill to be paid, usually by check, just like any other service.
This accounts for calling the tab a “bill,” a “check,” and for the hand-signing gesture. (Not necessarily completely; it was no doubt reinforced by other usages.) We still see vestiges of it when you stay at a hotel and sign the check to your room number and to the later practice of running a tab at a bar.