The call from Obama came after supporters of Dr Abdullah announced that he has won the presidential election and called for a cabinet formation.
The preliminary results of the presidential elections declared by the Independent Election Commission (IEC) of Afghanistan showed Dr Abdullah trailing by a million votes against his rival Ashraf Ghani.
Alleging massive electoral fraud, Dr Abdullah has refused to accept the election results.
The phone call by Obama to Abdullah is seen as a measure to prevent the situation in Afghanistan from going out of control.
"We've been clear that any such move would cost Afghanistan the financial and security assistance of the United States of America," Earnest said.
"Serious allegations of fraud have been raised, but they have yet to be adequately investigated. So we reiterate that the preliminary results that were announced yesterday are neither final nor authoritative and may not even predict the final outcome, which could still change based on the findings of Afghans' electoral bodies," he said.
Earlier, Secretary of State John Kerry expressed grave concern over reports of protests and of suggestions of a parallel government.
"Any action to take power by extra-legal means will cost Afghanistan the financial and security support of the United States and the international community," Kerry warned.
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