His comments came after Mick Huckabee accused Mr Obama of marching Israelis “to the door of the oven” by signing a nuclear agreement with Iran.
He said Mr Huckabee was trying to “get attention” after weeks of his rival Donald Trump dominating the debate.
Mr Obama also hit out at Mr Trump for the tone of his presidential campaign.
Speaking in Ethiopia in the middle of a short Africa tour, the president said attention-grabbing attacks “have become all too commonplace” in the US.
On Saturday, former Arkansas governor Mr Huckabee described Mr Obama’s foreign policy as “the most feckless in American history” and said the deal with Iran was “idiotic”.
The agreement, sealed on 14 July, places curbs on Iran’s nuclear programme in exchange for relief from US and international sanctions, ending more than a decade of tensions.
Mr Obama dismissed the comments but said they were part of “a general pattern” of unnecessary attacks, adding: “It’s not the kind of leadership that’s needed for America right now.”
The head of the Democratic National Committee said Mr Huckabee should apologise to the Jewish community.
But on Monday, he refused to back down and said he would stand with Israel “to prevent the terrorists in Tehran from achieving their own stated goal of another Holocaust”.
President Obama also took aim at Mr Trump, whose brash rhetoric has led the agenda for several weeks – and sent him to the top of several recent polls.
One of the billionaire mogul’s most eye-catching attacks was on John McCain, when he questioned the Republican senator’s status as a “war hero” because he was captured during the Vietnam war.
But Mr Obama said the comments were “outrageous” and Mr McCain was “somebody who endured torture and conducted himself with exemplary patriotism”.
“These are leaders in the Republican Party,” he said. “The American people deserve better. Certainly presidential debates deserve better.”
The Republican contest has grown increasingly bitter in recent days as the 16 candidates vying for the party’s nomination struggle to get their message across.
Former Texas Governor Rick Perry has labelled Mr Trump a “cancer on conservatism” while South Carolina Senator Lindsey Graham dubbed the billionaire businessman a “jackass”.
Only the top 10 Republican candidates will be included in the first televised debate on 6 August in Ohio. Their ranking will be drawn from an average of national polls.
The early Republican frontrunner was Jeb Bush, but he has been overtaken in some polls by Donald Trump.
Hillary Clinton will have learnt much from her failed campaign of 2008.
Florida Senator Marco Rubio lost some right-wing fans by backing a bipartisan immigration reform package.
Wisconsin Governor Scott Walker appeals to both the Republican establishment and the Tea Party.
Libertarian Rand Paul has his supporters – and enemies – among Republicans.
Veteran congressman Bernie Sanders is running as a Democrat despite never formally being part of the party.