Yes, it did come from the English spoken in England (and in Wales, Scotland, and Ireland) at the time that the first colonists left England. It wasn’t until after the time that the first American colonists left England that the r started to change into the way it’s presently spoken in England. American pronunciation of the r sound is thus closer to English pronunciation of it four hundred years ago than present British pronunciation is.
panache45 is talking about the way that r sounds in the middle or end of a word, not at the beginning where there isn’t much different between standard American pronunciation of r and standard British pronunciation of it.
cmyk, you’re not wrong, exactly. The distinctive American [r] is also still found in some British dialects, notably those in the southwest (Bristol, Plymoth, etc.). Which is where most of the British shipping industry was. Which is where most of the sailors and, yes, English-speaking pirates came from or spent time in.
This is why it’s especially funny when Americans go “Arrr, matey!” to sound like pirates. We’re making fun of a quirk of speech that we also have (most of us, anyway).
(1) As Elendil’s Heir points out, non-rhotic accents (as is RP) still pronounce the /r/ sound, just not under certain circumstances. I assume it is the non-rhoticity of RP that leads you to hazard that General American rhoticity (some American accents are non-rhotic) could not come from the British. It should also be pointed out that there are rhotic accents today in England.
(2) The American English/British English split antedates the development of non-rhoticity in general British English. Thus, the American /r/ did come from the Brits.
(3) Finally, and most importantly, the question presupposes that changes in language must come from some outside source in a clearly related way (i.e., if a language undergoes change X, then that change must have been generated by feature X in an outside influential language). There is no justification for this. For instance, nobody asks “Where did certain dialects of American English get the Northern Cities Vowel Shift from?” The principle of sufficient reason is a seductive notion, but it is a trap! It has no place in discussing undirected, stochastic but prodcutive-in-the-aggregate phenomena as language change or biological evolution (if only all the SDMB who asked “What’s the evolutionary reason for …?” would remember this salient fact!).
Yes, the R may be used differently in American and British dialects, but it’s basically the same sound, isn’t it, the Alveolar approximant? The various other R sounds that occur in Spanish, French, German etc. are truly different sounds, articulated differently in the mouth.
I’m curious to know more about this accent - is there a name for it? I grew up near Cleveland and I never noticed that the area had anything other than the standard broadcaster accent. I’ve been told once that I have an “Ohio accent” and I have no idea what that means. I sound like newscasters as far as I know - IOW no discernable accent aside from the “standard” American one.
The rolled [r] is still found in some English dialects (Scotland, parts of Wales) and some speech registers (traditional theater, and if memory serves some of the British upper class). In dialects with no post-vocalic [r] (standard British speech, parts of the South, parts of the Northeast), vowel + r has two possibilities: VR > V (as in New England) or VR > rhotacized vowel (as in England). British speakers I know can easily distinguish between pawn and porn in their dialect by the quality of the vowel, whereas they sound identical to my ear, since in my dialect the -aw- and short -o- are homonymous while post-vocalic [r] is the same as initial [r]. This would be a lot easier if I could type IPA. And of course French speakers who have learned English often continue to use a uvular [r], etc.
This is recent. I was born and raised in the Cleveland area, then lived in New York for 25 years, then returned here. Current speech in Cleveland is subtly different than it was in my youth, I think due to influence from the African-American community. The only way I can describe it is that there’s a small amount of “twang” that didn’t used to exist. “BEEyer” is a good example. I never used to “HEEyer” that.
Note that this is not an Ohio accent. It affects an area of the U.S. that includes part of Ohio. It doesn’t affect most of Ohio. Interestingly, although it comes right up to the Canadian border, it hasn’t crossed into Canada.
Not only that, we seem to have a hard time even hearing some sounds that aren’t part of our native language. I was talking with a Venezuelan coworker of mine, and asked him the difference between the “v” sound and the “b” sound in Venezuelan Spanish: