Hours after the video was released Thursday, Al-Qaeda in the Arabian Peninsula, claimed a rocket attack on the "American section" of Al-Anad air base in the southern province of Lahij, according to US-based monitor SITE Intelligence Group.
Yemen has acknowledged that US personnel gathering intelligence for drone strikes on AQAP are deployed at Al-Anad, which a military official confirmed was attacked Thursday.
AQAP said the attack was in revenge for the "killing of Muslims" during a failed military operation to rescue US journalist Luke Somers, SITE reported.
Somers and South African teacher Pierre Korkie were shot and fatally wounded by their captors Saturday when American commandos stormed the Al-Qaeda hideout where they were being held in southeastern Yemen, US officials said.
"Obama made the wrong decision, and signed the death warrant" of Somers, AQAP official Nasser bin Ali al-Ansi said in the video.
The raid came two days after the kidnappers issued a video in which they threatened to kill Somers within 72 hours unless Washington met unspecified demands.
Ansi said Obama had ordered the assault "despite our warning that he should not do anything foolish".
The failed operation came just a day before Korkie, 57, was due to be freed under a deal with the kidnappers, although the US ambassador to South Africa has said Washington was unaware of his impending release.
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"Obama made a decision that caused things to take a completely different turn to what we wanted," Ansi said.
"He could have avoided the fate of this hostage, which was the inevitable consequence of the military solution," he added.
Ansi said AQAP was seeking the release of detainees held by the United States, including blind Egyptian sheikh Omar Abdul Rahman and Pakistani Aafia Siddiqui, jailed on terrorism charges.
Denouncing US policy towards the "Muslim nation," Ansi warned that "the lives of all Americans are in danger, in or outside the United States, in the air, on land and at sea."
"Can your nation dream of security when it lives in such a situation? You cannot dream of security until security is a reality in Palestine and all Muslim countries."
Somers was seized in Sanaa in September 2013. Korkie was abducted together with his wife in May 2013, but she was released in January.
AQAP, considered by Washington as Al-Qaeda's most dangerous affiliate, has exploited instability in the impoverished country since a 2011 uprising forced president Ali Abdullah Saleh to step down.
In recent years it has carried out a growing number of abductions, with several hostages still in captivity.
Yemen is a key ally of the United States, allowing Washington to carry out a longstanding drone war on its territory against AQAP.