Preliminary results showed Ashraf Ghani, a former World Bank official, won the June 14 second round, but his rival Abdullah Abdullah immediately rejected the outcome, saying the vote was marred by widespread fraud.
Underscoring the magnitude of the crisis, Mr Abdullah said US Secretary of State John Kerry would visit Kabul on Friday.
And the politiical crisis came as at least 16 people, including four Czech Nato soldiers, were killed in a Taliban suicide attack near a clinic in eastern Afghanistan.
Thousands of Abdullah supporters gathered in Kabul yesterday, demanding their leader form a parallel cabinet and unilaterally assert his own rule - a dangerous move that would further fracture the fragile country.
Mr John Kerry said there was no justification for violence or "extra-constitutional measures".
He added: "I have noted reports of protests in Afghanistan and of suggestions of a 'parallel government' with the gravest concern.
"Any action to take power by extra-legal means will cost Afghanistan the financial and security support of the US and the international community."
Afghanistan is heavily reliant on foreign donors to fund everything from building roads and paying school teachers to security. The US pays the lion's share of all international aid.
Observers fear a stand-off between Mr Abdullah and Mr Ghani could plunge Afghanistan into disorder, with no clear leader in a country already beset by deep-rooted ethnic divisions.
Mr Abdullah has accused President Hamid Karzai, who is stepping down after 12 years in power, of helping rig the vote in favour of Mr Ghani, describing it as a "coup" against the people.
The stand-off over the vote has quashed hopes for a smooth transition of power in Afghanistan, a concern for the West as most US-led forces withdraw from the country this year. There are concerns, however, about how much Mr Abdullah, who is popular among the powerful Tajik community in the north, would be able to control his supporters if the crisis escalated.
At the rally in Kabul, Mr Abdullah, visibly flustered by the size of the gathering, faced a roar of slogans demanding he immediately announce his own cabinet, telling supporters to be patient.
He said: "We are the winner of this round of elections without any doubt. The people of Afghanistan call on me to announce my government today.
"We cannot ignore this call. Once again I ask you to give me a few days days to consult and speak."
Away from the city's centre, dozens of Abdullah supporters tore down a Karzai portrait at Kabul's international airport.
The Independent Election Commission announced on Monday Mr Ghani had won the second round with 56.44 per cent of the vote, according to preliminary results. The tally might change when the final official numbers come out on July 22.
Mr Ghani is backed mainly by Pashtun tribes in the south and east of the country.