Defending Mr Johnson, Coun Edwards said he thought there were "much more important things to worry about".
Coun Edwards, who represents Morecambe South and is cabinet member for highways and transport, said: "In the middle of an unprecedented pandemic every single person you speak to will have broken a rule in some way."
He said Mr Johnson was "ready to fight on and ready to take on board what has happened".
"Now is the time for No 10 to be absolutely reorganised," he added. "I was always against lockdowns and now is the time to absolutely move on."
Coun Edwards later added on social media: "I also said that I welcome the apology he made, accept he needs to make changes, surround himself with Conservatives in Number 10 again and that there are much more important things to worry about: Russia, cost of living, levelling up, vaccines."
However, following complaints about the comments, Coun Edwards has now retracted what he said with an apology on social media.
"I used a really poor choice of words in the interview on Monday and wish to retract my comment about how people behaved during the lockdowns," he said.
"Of course, people followed the rules and I sincerely apologise for the offence and any hurt I have caused.
"This is not a trivial matter, and everyone is right to be angry. So many people made so many sacrifices. Especially those who have been working hard to support those recovering from the pandemic.
"I have apologised privately to my colleagues on the council and I want to offer a full and unreserved apology to the people of Lancashire whom I serve."
The Metropolitan Police is currently investigating 12 potential breaches of Covid-19 laws at No 10, with Ms Gray's full report due following the conclusion of these proceedings.This morning,