The member for St Austell and Newquay, who previously worked in his local church, released a statement amid newspaper speculation over his marriage.
In the statement, he said: "I have behaved completely inappropriately and deeply regret the pain I have caused those closest to me," he said.
"Anne and I are committed to being together and seeking to move forward in our marriage.
"I accept I have work to do to recover the trust and respect of those around me. I would respectfully request that people give us the time and space we need at this time."
Mr Double has been married to wife Anne for 29 years and they have three sons.
He became an MP in 2015 taking the seat from the Liberal Democrats.