Most North American speech is rhotic—why is that? Does it come from the early English settlers or perhaps from the Irish settlers?
As reported on Wikipedia (Rhotic and non-rhotic accents), English had become non-rhotic by the end of the 18th century; John Walker used the spelling ar for the pronunciation of aunt in 1775, and reported caad as pronunciation of card in 1791.
British colonization of the Americas began in 1607 in Virginia, even though there had been previous attempts in 1586 and 1587. United States of America declared its independence on July 4 1776, when the non-rhotic accent had started to spread on Southern England.
As kiamlaluno noted in his comment, it's a question of when predominant English dialects transitioned from rhotic to non-rhotic. While I don't have any definite (or academic) reference to supply, three comments on this:
One can read here: “In 1776, both American accents and British accents were largely rhotic. It was around this time that non-rhotic speech took off in southern England, especially among the upper class. This “prestige” non-rhotic speech was standardized, and has been spreading in Britain ever since.” The author offers as a reference The Cambridge History of the English Language: English in North America. I located the correct passage at page 75 and later.
This datation is backed by the following Wikipedia quote: “Loss of coda /r/ apparently became widespread in southern England during the 18th century; John Walker uses the spelling ar to indicate the broad A of aunt in his 1775 dictionary and reports that card is pronounced "caad" in 1791 (Labov, Ash, and Boberg 2006: 47).”
Even by 1950, large parts of England still had largely rhotic dialects (see maps here).
Expanding a little on the info above: The perception that American English and British English somehow developed largely independently after North America was colonized is not correct. Non-rhoticity started around London, then gradually spread northward and westward around England, and then spread to North America. In North America it started on the East and Gulf coasts and then spread inward. By the early 20th century American culture was starting to really assert itself and Americans stopped looking to England for what was fashionable. As the Midwestern and Western states became economically more important, the Midwestern (rhotic) accent became established as the "standard" accent, especially with the advent of radio and television. The accent trend began to reverse and rhoticism re-asserted itself eastward in North America.
But it is important to note that, even today, New York and New England still have large numbers of non-rhotic speakers, as do New Orleans and some other parts of the South. Similarly southwestern England is still heavily rhotic as are Ireland and Scotland.
Here is an article from the BBC on this subject. How Americans preserved British English:
One feature of most American English is what linguists call ‘rhoticity’, or the pronunciation of ‘r’ in words like ‘card’ and ‘water’. It turns out that Brits in the 1600s, like modern-day Americans, largely pronounced all their Rs. Marisa Brook researches language variation at Canada’s University of Victoria. “Many of those immigrants came from parts of the British Isles where non-rhoticity hadn’t yet spread,” she says of the early colonists. “The change towards standard non-rhoticity in southern England was just beginning at the time the colonies became the United States.”
One reason I heard for Britain becoming non-rhotic that made sense is that in the early 1800s the British royals married with German aristocracy. German was non-rhotic, and the imported royals had a non-rhotic accent (think “Ahnold Swarzeneggah” type accent) So the British upper classes were quick to start copying the new “posh” accent, and it spread from there.
French émigrés (of 1786-1815, Wikipedia) brought to London their habit of gobbling the last letter of every word (unless a vowel followed). Locals could thus affect an aristocrat's accent by gobbling the terminal "r". The softening of the interpretation of the spelled "r" may have been enough to divert the evolving accents.