In a message to Bush, the Yemeni head of state said, "Those who committed this ugly crime will not evade punishment."
Saleh expressed deep regret for the killings and condemned what he called "this criminal action which contradicts the beliefs and values of the Yemeni people."
President Bush was informed of the shootings by his staff early Monday morning at his Texas ranch, where he is spending the New Year's holiday. The White House strongly condemned the attacks.
"We are working closely with Yemeni officials to investigate these attacks," the White House said. "Yemeni officials already have one individual in custody, and we will work with them to bring to justice all those who are responsible."
The three missionaries -- all health workers -- were shot and killed by a lone gunman who, according to Yemeni authorities, entered the room where they were attending a meeting and opened fire. A fourth American was wounded in the attack at Jibla Baptist Hospital, and his condition was later said to be serious.
Yemeni security authorities said later they had apprehended Abid Abdulrazzaq Al Kamil, 35, in connection with the shooting. Police said he was a suspected Islamist militant. He was a student at Yemen's al Iman university, which was briefly closed last year after allegations that it was a hotbed of Islamist militancy.
The dead staff was identified as the hospital's administrator, William Koehn, 60, of Arlington ,Texas; obstetrician, Dr. Martha Miers, 57, of Montgomery, Ala., and 53-year-old Kathleen Gariety, the hospital's purchasing agent, who comes from Wauwatosa, Wis. Pharmacist Donald Caswell, underwent surgery to remove bullets from his abdomen.
The hospital is located in the town of Jibla, in Ibb province, about 100 miles south of Sanaa.
A hospital source spoke of their long service in the country.
"They have been serving Yemen for 20 years and were about to leave after a transition of administration was going to apply," the source said.
Sources at the hospital said the attack appeared to have been planned in advance. The killer might also have known of the planned meeting to discuss the transition of the administration of the humanitarian services in the hospital to a Yemeni non-governmental organization or a British association.
Islamist militants, including Osama bin Laden's al Qaida organization, have long found a safe haven in the Yemen, where a weak central government has little control over the tribal areas outside the main cities.
Last month, an unmanned aircraft operated by CIA agents targeted and killed six al Qaida suspects in the country.
A statement from the U.S. embassy in Sanaa called on U.S. citizens to enhance their security.
(Hussain Hindawi in London contributed to this report)