Both have denied the charges and have said they will cooperate with the court to clear their names.
The assurance has not allayed fears among some Western diplomats.
Before the election, Johnnie Carson, the State Department's top diplomat for Africa, issued a stern warning, saying Kenyans can pick their own leaders, but "choices have consequences."
But in a statement after the election, U.S. Secretary of State John Kerry applauded the vote.
"Kenya has been one of America's strongest and most enduring partners in Africa," Kerry said. "We ... will continue to be a strong friend and ally of the Kenyan people."
The UK minister for Africa, Marks Simmonds, echoed his sentiment and called on candidates to address disputes in court to avoid violence.
Kenyatta has said the indictment will not affect his ability to do his job and urged the international community to respect the will of Kenyans.
"I have been a member of the government, and I've still been able to execute my duties," said Kenyatta, a deputy prime minister in the outgoing administration. "I still believe I will be able to execute my duties as president."
Carson's predecessor, Jendayi Frazer, said she does not foresee much change in ties between Kenya and the United States.
"Kenyatta knows that he needs the United States, and the United States knows it needs Kenya," she said. "And so I suspect that while it might be awkward, there won't be a significant change in our policy stances toward Kenya or theirs toward us."
No faith in the ICC
Analysts say the ICC indictment may have rallied citizens to Kenyatta's side in defiance of the West.
"Many Africans have lost faith in ICC and view it as targeting African leaders and failing to discharge its justice among non-African leaders," said Ayo Johnson, director of ViewPoint Africa. "Kenya sent a loud message to the ICC ... don't interfere. And it does not matter if you brand our leaders as criminals."
The president-elect's trial is scheduled for July while his running mate's is in May.
Kenya is the second African nation after Sudan to have a sitting president facing charges at the International Criminal Court.
Eager to avoid a repeat
In a nation with fierce ethnic loyalties, Kenyans were eager to avoid a repeat of the last election.
At the time, the nation plunged into ethnic violence after Odinga disputed results that declared the incumbent president, Mwai Kibaki, as the winner, alleging the election had been rigged.
Ethnic groups loyal to each side took to the streets, where supporters of both camps fought one another. More than 1,200 people were killed and hundreds of thousands displaced.
But despite the divisive vote this year, some citizens said it is time to move forward.
"The people have spoken, even though my voice was not one of them," said Jane Nduku, a Nairobi resident.
Kenyatta's win at age 51 makes him the nation's youngest president.