The GOP has been aggressive on social issues in many states since the 2010 election, The New York Times reported.
John Weaver, a strategist for Sen. John McCain of Arizona during the 2008 presidential race and for former Utah Gov. Jon Huntsman Jr. this year, said that could hurt the Republican effort to win voters in some significant groups, including independents, young people and women.
"I think it's problematic not just for this national election we're facing, but for the long-term health of the party," Weaver said.
Indiana Gov. Mitch Daniels, on the other hand, suggested the party and its presumptive nominee, Mitt Romney, can win this year by focusing on economics in the national campaign and leaving the social-issue fights to the states.
Some governors are finding social issues tough to leave behind when they make national appearances. Speaking on MSNBC's "Morning Joe" as head of the Republican Governors Association, Virginia Gov. Bob McDonnell said he wanted to talk about the drop in unemployment in his state but also had to answer questions about his decision to sign a bill requiring ultrasounds before women can get abortions.