Nabil Amr, a top aide to Abbas in the West Bank, praised the efforts of Qatar and accused Hamas of stalling.
"We regret the failure of this initiative," he said. "They are buying time and time is of the essence for us."
Amr said the president is now weighing other options for resolving the standoff, including replacing the Hamas-dominated Cabinet with a government of apolitical technocrats or taking steps to hold early elections.
"I don't think there is room for any more international initiatives," he said.
Qatar has emerged as a possible mediator in the standoff between Israel and Hamas. The Gulf state has low-level relations with Israel and has good relations with Hamas' exiled leadership, which is based in Syria.
Sheik Hamad did not comment before his departure. But after meetings on Monday night, he said only that his country's plan was the center of discussions, and there was still no agreement on the issue of recognizing Israel.
Hamas on Tuesday claimed it had fired a homemade rocket at Israeli troops in northern Gaza. The army said militants fired an anti-tank missile at troops, causing no injury or damage.
"Resistance is the only way to deter [Israel's] aggression," Hamas said in a statement. "Seeking flexibility, dialogue and negotiations will not do any good."
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