NASHUA, N.H.—The biggest gathering yet of Republican presidential hopefuls this past weekend sharpened divisions in the broad field of candidates over the path to return the GOP to the White House, making the 2016 primary race a moment for the party to define its national identity.

On one side are Jeb Bush and Sens. Rand Paul and Lindsey Graham, who emphasized that the party’s nominee needs to reach beyond the GOP’s base of mostly white and older conservative voters, given how the general electorate is growing increasingly diverse.

On the other sits a handful of presidential contenders, including Sen. Ted Cruz and Louisiana Gov. Bobby Jindal, who say the party’s nominee needs to be a dedicated conservative who can mobilize the religious right to turn out in droves in 2016.

Republicans in recent years have wrested control of Congress from Democrats, decisively in the case of the House, and they control a plurality of state governments. But President Barack Obama’s two wins showed how a Democrat can win nationally through a diverse coalition of voters without relying on traditional Republican strongholds.

Party leaders say Republicans can’t afford to choose between the two approaches: The party’s nominee has to both excite grass-roots conservatives and tap slices of the electorate that don’t lean traditionally Republican.