In an apparent reference to Partygate in a letter to Boris Johnson, Mr Dowden said he shared the feelings of Tory supporters who were “distressed and disappointed by recent events”.
But the senior MP stopped short of criticising the PM, calling his decision “deeply personal” and suggesting he bore some responsibility for a “very poor run” of results.
“Yesterday’s parliamentary by-elections are the latest in a run of very poor result for our party,” Mr Dowden wrote. “Our supporters are distressed and disappointed by recent events, and I share their feelings.”
He added: “We cannot carry on with business as usual. Somebody must take responsibility and I have concluded that, in these circumstances, it would not be right for me to remain in office.”
Senior Tory MP Simon Hoare – who voted against Mr Johnson at the recent confidence vote – said Mr Dowden was “not to blame” for the losses. “An honourable letter from an honourable man,” he tweeted.
Fellow Tory rebel Andrew Bowie said he was “really sorry” to see the “thoroughly decent man” depart the role.
The Liberal Democrats swept to victory in Devon, where Richard Foord overcame a Tory majority of 24,239 votes and won by more than 6,000 – the largest ever majority overturned at a by-election.
Lib Dem leader Sir Ed Davey said the result should be a “wake-up call” to Tory MPs about Mr Johnson’s leadership, while Mr Foord called on Mr Johnson to “go now”.
Labour’s Simon Robert Lightwood won in Wakefield after winning the west Yorkshire seat by almost 5,000 votes, overturning a smaller Conservative lead of 3,358 votes.
Labour leader Sir Keir Starmer said the result showed that the country has “lost confidence in the Tories” and that his party was “ready for government”.
The double defeat will raise further questions for PM, currently in Rwanda for the Commonwealth summit, who insisted on Thursday that it would be “crazy” to step down even if he lost both by-elections.
Former No 10 adviser Tim Montgomerie, creator of ConservativeHome website, said the “massive” result in Tiverton showed why Mr Johnson had to be replaced. “This is a crisis for the Conservative party,” he told Sky News.
He added: “Voters are reacting to the character of the prime minister. They are rejecting the character of the prime minister. And if the Conservatives don’t act soon … the whole Conservative party will be judged. We cannot let this situation continue.”
Former Tory special adviser Mo Hussein said there would have to be a lot of “soul searching” about how to move forward – suggesting the party should ditch its emphasis on “culture war” issues.
Mr Foord, who becomes the first non-Tory MP to win Tiverton since its creation in 1997, also warned the PM that “coups can happen” when leaders head overseas during rocky patches at home.