Ugandan police said a Briton and a South African were killed along with their guide in Tuesday's attack, which has been blamed on an armed militia based in the neighboring Democratic Republic of Congo.
The trio were targeted by gunmen as they were on safari in Queen Elizabeth National Park in southwestern Uganda and their vehicle set ablaze, police and park officials said.
"It was a cowardly act on the part of the terrorists attacking innocent civilians and tragic for the couple who were newlyweds and visiting Uganda on their honeymoon," Museveni said on X, formerly Twitter.
"Of course, these terrorists will pay with their own wretched lives."
In the aftermath of the killings, the British government issued a warning to its citizens against travelling to the popular park, a tourist magnet where lions are known for their unusual ability to climb trees.
Police blamed the attack on the Allied Democratic Forces (ADF) rebel militia that is affiliated to the so-called Islamic State group.
- 'Terrorists will be defeated' -
Museveni called on Uganda's army and other security forces to ensure that "these mistakes do not happen again and that the ADF is wiped out."
"The terrorists will be defeated like (Joseph) Kony was defeated," he added, referring to the fugitive leader of the notorious Lord's Resistance Army, which terrorized parts of central Africa for more than three decades.
Tuesday's attack came two days after Museveni said police had foiled a bomb plot by the ADF on churches about 50 kilometers (30 miles) from the capital Kampala.
Museveni said on Sunday that the ADF had made two bombs but that they "were reported to police and defused."
Earlier that day, the veteran leader also said Ugandan forces had carried out a series of air strikes against ADF positions in DRC and claimed that a number of the militants were killed.
He warned that "the terrorists are running from Congo... and re-entering Uganda and trying to commit some random terrorist acts."
Queen Elizabeth Park shares a border with DRC and its renowned Virunga National Park, a habitat for rare mountain gorillas, and where armed groups are believed to operate.
In 2019, a U.S. tourist and her safari guide were kidnapped by four gunmen during an evening game drive through the Ugandan park. They were recovered unharmed after a ransom was paid.
The ADF is historically a Ugandan rebel coalition whose biggest group comprised Muslims opposed to Museveni.
Established in eastern DRC in 1995, the group became the deadliest of scores of outlawed forces in the deeply troubled region.
It has been blamed for massacres, kidnappings and looting, with a death toll estimated in the thousands.
In June, ADF fighters killed 42 people including 37 students in a high school in western Uganda near the border with DRC.
Tourism is a top foreign exchange earner in Uganda, contributing almost 10 percent of GDP last year, according to government figures.
Uganda's wildlife officials said all parks will remain open despite Tuesday's attack.