HAVANA — President Obama stood beside President Raúl Castro on Monday and declared a “new day” of openness between the United States and Cuba, but old grievances and disputes over human rights marred a groundbreaking meeting and underscored lingering impediments to a historic thaw.

The two presidents, meeting at the Revolutionary Palace for the first such official contact between their two governments in more than a half-century, engaged in a frank and at times awkward exchange with each other and reporters. Mr. Obama at turns prodded Mr. Castro to submit to questions during an extraordinary 55-minute news conference.

Standing at lecterns in a cavernous granite-walled hall in front of Cuban and American flags, the two leaders traded criticism of each other’s countries even as both said they were committed to continuing on the path to normalizing relations.