INDIANAPOLIS — A conservative think tank recently sent Ben Carson, a potential presidential candidate and fierce critic of the federal health care overhaul, a series of messages to post on Twitter during the grilling in the House of Jonathan Gruber, the M.I.T. economist who advised the Obama administration.

Mr. Carson, a retired pediatric neurosurgeon, has set off a prairie fire of rank-and-file conservatives urging him to run for the Republican nomination. He liked the think tank’s idea, but he knows that he excites supporters because he is an unscripted outsider. So he wrote his own Twitter posts, including, “Why was it necessary to disrupt entire medical system to take care of the needs of 15% of the people?”

With former Gov. Jeb Bush of Florida moving closer to declaring his candidacy, and the Republican establishment determined not to repeat the mistakes of the lengthy nominating fight of 2012, the rise of Mr. Carson suggests the best-laid plans of the party for 2016 may not go as smoothly as desired.

Even as Mr. Bush began to quietly line up donors, and the attention of the news media turned last week to policy disputes over Cuba between two other potential high-profile contenders, Senators Marco Rubio and Rand Paul, Mr. Carson was riding an under-the-radar groundswell.