"I want to extend my apologies to the people for not being able to complete my duties," he said in a statement read to reporters following the cabinet's final meeting.
Abe, 53, stunned the nation when he announced his resignation on September 12.
He had faced harsh criticism for abruptly resigning amid a parliamentary battle over the extension of the country's naval mission in support of US-led troops in Afghanistan.
On Monday however he said that he had resigned for unspecified health reasons.
Naval mission
Abe has been in hospital since resigning
on September 12 [Reuters]
Ahead of Tuesday's parliamentary vote Fukuda met Akihiro Ota, the head of the LDP's New Komei ruling coalition partner, to reaffirm the parties' solidarity.
But despite being assured of the prime ministership Fukuda is unlikely to get an easy ride once in office.
His first order of business will be pushing the extension of the Indian ocean naval mission through parliament, where the opposition has vowed to defeat it.
Japanese tankers have been refuelling coalition ships in the Indian ocean since 2001, and the US – Japan's top ally and protector - has been pushing for an extension of the operation.
With the LDP's reputation in tatters and the upper house of parliament dominated by the opposition, calls are growing for snap elections that could endanger the party's grip on the lower house.