The pontiff waved to crowds of hundreds of supporters at the airport, many from local Catholic parishes, and shook the US president's hand before leaving for the Vatican embassy in the city.
He is to meet the US president again on Wednesday when he greets a crowd of supporters at the White House for his birthday.
Benedict and Bush have disagreed on a number of issues, most notably over the war in Iraq and the US embargo on Cuba, but have found common ground in opposing abortion, gay marriage and stem cell research.
A poll released this month by the Pew Forum on Religion and Public Life showed the German-born pope was viewed favourably by most people in the US, but was not as popular as his predecessor, John Paul II.
Call for action
"We're way beyond the point at which an apology, a nice gesture, a few soothing words and promises, will be meaningful"
Survivors Network for those Abused by Priests
The trip is the first by a Roman Catholic pope since a wave of abuse scandals began in the US in 2002, provoking legal actions that led to more than $2bn in settlements.
Benedict said the Catholic Church would do everything possible to screen candidates for the priesthood "so that only really sound persons can be admitted".
"It is more important to have good priests than to have many priests," he said.
However, he is not expected to meet abused victims of paedophile priests, and one group of victims, the Survivors Network for those Abused by Priests (Snap), said more had to be done.
"We're way beyond the point at which an apology, a nice gesture, a few soothing words and promises, will be meaningful," Snap said in a statement quoted by AFP.
Priesthood in decline
Benedict's visit also comes amid concerns over a decline in the number of American men joining the priesthood.
A recent Catholic University study found that 17 per cent of priests serving in American churches come from other countries.
Enrolment in the four-year theology programmes required to join the priesthood has not risen for a decade, the New York Times reported.