WASHINGTON — The announcement Sunday by Gov. Mitch Daniels of Indiana that he would not run for the Republican presidential nomination ended one major chapter of uncertainty in the race but ignited new debate over whether the current field contains a candidate capable of beating President Obama next year.

Saying that his family did not want to go through a campaign, Mr. Daniels became the third high-profile Republican in eight days to choose not to compete for the chance to challenge Mr. Obama. The contest is now increasingly focused on three former governors — Mitt Romney of Massachusetts, Tim Pawlenty of Minnesota and, to a lesser extent, Jon M. Huntsman Jr. of Utah.

Mr. Daniels’s decision was one of the most anticipated events of the early campaign season, and he reached it just days after former Gov. Mike Huckabee of Arkansas and the developer Donald J. Trump said they would opt out of a race. Mr. Daniels faced considerable pressure from a wide spectrum of Republicans to enter the race, and he had signaled that he would focus on addressing the nation’s fiscal issues.

But after weeks of deliberating in public and making clear that his wife and four daughters had deep reservations — caused in part by the knowledge that they would be exposed to intensive scrutiny over a period in the 1990s when Mr. Daniels and his wife, Cheri, divorced and then remarried — he said he was unsuccessful in swaying his family.