Museveni flew into Juba, the capital of South Sudan, aboard a military helicopter on Monday, the day he had been scheduled to sign the peace deal.
"I have come prepared to sign the agreement but Kony has not showed up," he said.
"It is clear that Kony is the one who is not serious."
However, Kiir has said that peace talks are still on track despite the no-show by Kony.
"The negotiation has not ended," Kiir said.
"The fact that Joseph Kony has refused to sign the agreement does not mean the end of the road."
Twenty years of fighting have left tens of thousands dead and displaced two million people, mainly in northern Uganda.
Several thousands have also been killed in southern Sudan, where the LRA has camps.
A ceasefire was struck in August 2006, leading to a year and a half of negotiations up to the Juba talks.
Kony, a semi-literate former altar boy, took charge in 1988 of a regional rebellion among northern Uganda's ethnic Acholi minority.
Since the signing of a truce almost two years ago, the nation has engaged in aggressive reforms to attract investors and revamp its image.