Watson told the BBC that while he would not take on Corbyn, he expected there to be a challenge from among the party's MPs, adding that he had asked the leader if a "negotiated settlement" was possible, but found him "unwilling to move from his position".
"It looks like the the Labour Party is heading for some form of contested election," he said, adding of his own ambitions, "the prerequisite for being a deputy is you never want to be leader".
A challenge to the leader has long been predicted by party insiders, and Watson's refusal to stand may open the door for other contenders, which have been rumoured to include former shadow business secretary Angela Eagle.
It comes at the end of four days of crisis for the Labour party, with Prime Minister David Cameron earlier calling on Corbyn to step down "in the national interest".
The Labour party has now been hit by more than 60 resignations from its shadow cabinet, front bench and parliamentary aides, with many citing the party's ineffectiveness in opposition under Corbyn, who yesterday lost a non-binding vote of confidence among his MPs 172-40.
However, the party leader remains determined, and yesterday said that he would not "betray" party members who had elected him by resigning.