At the end of the third book Peeta does not kill Prim. Prim dies while trying to help children who were being used as human shields to protect President Snow from the rebel invasion. A bomb was detonated in an area of children, and Prim rushed in with a medical crew to try to save them. After a few minutes, Katniss watched in horror as a second bomb went off and killed her sister.
Who killed her, is a much deeper question that plagues Katniss throughout the end of the novel. At first her thoughts are that the hover craft bearing the emblems of the Capitol were under the orders of President Snow. Later she recalls that Gale was working on a two stage bomb similar to the one used to kill her sister and she begins to blame him, realizing that the emblems could have been put on the craft by the rebellion to stir up hatred against the Capitol. In the end she realizes that Coin, leader of the rebellion, would be responsible for authorizing a child of Primrose's age to be anywhere near a battle scene so she begins to blame Coin.
Ultimately, Collins leaves it to the reader to make up their own mind and share in the torment of Katniss as we weigh the options together. With that being said, Peeta was in no way responsible, and perhaps even saved Prim's life earlier when he warned District 13 of the impending attacks on the national television address.