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.