It was unclear when the video was recorded.
At the White House, spokesman Jay Carney insisted the United States would not negotiate with al-Qaeda. "We condemn his kidnapping in the strongest terms and call for [Weinstein's] immediate release," Carney said.
A woman who answered the phone Monday at a number listed for Weinstein in Rockville, Md., said she had no comment when an Associated Press reporter identified herself. Phone messages left for Weinstein's relatives were not immediately returned.
Weinstein was abducted in August in the eastern Pakistani city of Lahore after gunmen tricked his guards and broke into his home. He was the director in Pakistan for J.E. Austin Associates, a U.S. firm that advises a range of Pakistani business and government sectors.
"It's important you accept the demands and act quickly and don't delay," Weinstein said in the video, addressing Obama. "There'll be no benefit in delaying. It will just make things more difficult for me."
Weinstein spoke while sitting in front of a white background. He wore a white shalwar kameez, the loose-fitting clothing common in both Pakistan and Afghanistan. Several books and what appeared to be a plate of food were set in front of him. Weinstein took several bites of food as he spoke.
He appealed to Obama as a father. If the president responds to al-Qaeda's demands, Weinstein said, "then I will live and hopefully rejoin my family and also enjoy my children, my two daughters, like you enjoy your two daughters."
The video was released by Al-Sahab, al-Qaeda's media arm. It was first reported by the SITE Intelligence Group, which monitors messages from militants.
After his kidnapping, Weinstein's company said he was in poor health and provided a detailed list of medications, many of them for heart problems, that it implored the kidnappers to give him.
In the video released Sunday, Weinstein said he would like his wife, Elaine, to know "I'm fine, I'm well, I'm getting all my medications, I'm being taken care of."
Mike Redwood, a friend of Weinstein's from Somerset, England, said he watched the video Monday morning and had mixed feelings. He said he was grateful that Weinstein is alive - or at least was alive when the undated video was shot - but remained dismayed to see his friend in such dire circumstances.
"He's more capable of withstanding these circumstances than anybody else I know," Redwood said. "But it doesn't take away from feeling really depressed at seeing him there."
Redwood said he hoped he could take Weinstein at his word that he was getting his medications and being treated well. He said the poor image quality of the video made it difficult to gauge his health.
Redwood, a leather industry consultant, met Weinstein when they worked together on a plan to enhance the Pakistani leather industry. He has not spoken to Weinstein's family.