Defense Department officials balked. While they did not deny some American culpability in the episode, they said expressions of remorse offered by senior department officials and Secretary of State Hillary Rodham Clinton were enough, at least until the completion of a United States military investigation establishing what went wrong.

Some administration aides also worried that if Mr. Obama were to overrule the military and apologize to Pakistan, such a step could become fodder for his Republican opponents in the presidential campaign, according to several officials who declined to be named because they were not authorized to speak publicly…

With everything at stake in the relationship with Pakistan, which the United States sees as vital as it plans to exit from Afghanistan, some former Obama administration officials said the president should make public remarks on the border episode, including a formal apology.

“Without some effective measures of defusing this issue, Pakistan will cooperate less rather than more with us, and we won’t be able to achieve our goals in Afghanistan,” said Vali Nasr, a former State Department official who specialized in Pakistan.