But the Foreign Secretary stopped short of meeting repeated requests to admit he made a mistake and offer an unequivocal apology.
Mr Johnson also accused Labour of political point-scoring and deflecting blame from the Iranian regime over the imprisonment of Nazanin Zaghari-Ratcliffe
The Cabinet minister faced a grilling from MPs after last week saying Ms Zaghari-Ratcliffe was training journalists in Iran at the time of her arrest last year, something her employer and her family insist is incorrect.
Speaking in the Commons, Liberal Democrat Layla Moran said two of her Oxford West and Abingdon constituents are family members of Ms Zaghari-Ratcliffe.
She said of Mr Johnson: “The family has been worried sick by his irresponsible comments.
“On a human level is the Foreign Secretary at all sorry for the rollercoaster of emotions he has caused Nazanin and her family this week, and could he at least apologise to them today?”
Mr Johnson replied: “Of course I’m sorry if any words of mine have been so taken out of context and so misconstrued as to cause any kind of anxiety for the family of Nazanin Zaghari-Ratcliffe, of course I am.
“But the most important thing is that I don’t believe, and I have this from the Iranians themselves, that those words had any impact on the judicial process.
“We are going to work flat out to secure the release of Nazanin Zaghari-Ratcliffe and I’m very happy to have been able to make that clarification to the House today and I’m delighted, as I say Richard Ratcliffe welcomes the clarification I have made.
“But if she passes on my thoughts to her constituents, who are the family of Nazanin Zaghari-Ratcliffe, I’d be very grateful.”
Shadow foreign shadow secretary Emily Thornberry said Mr Johnson should show some personal responsibility and stand down as Foreign Secretary.
She also said Mr Johnson should apologise to Ms Zaghari-Ratcliffe’s family, friends, employers and everyone else “who have been working so hard to obtain this young mother’s release for the distress and anguish that his foolish words have caused to them and to this woman in recent days”.
Ms Thornberry added: “How many more time does this need to happen? How many times does the Foreign Secretary have to insult our international partners, damage our diplomatic relations, and now imperil the interests of British nationals abroad?
“What will it take before the Prime Minister says enough is enough?”
Labour’s Tulip Siddiq (Hampstead and Kilburn), Ms Zaghari-Ratcliffe’s MP, also urged Mr Johnson to apologise and retract his comments made to the Foreign Affairs Select Committee.
Labour former minister Chris Bryant, a member of the committee, said: “There’s not a single eight-year-old in the country who couldn’t say to the Foreign Secretary this does not match what you said last week.
“There’s not a single eight-year-old that wouldn’t be able to tell the Foreign Secretary how to do his job better.
“The honest truth is if you can’t show some contrition today then the honest truth is he shouldn’t be in his job as our people aren’t safe.”
Mr Johnson reiterated he had already answered the point.
Wes Streeting, Labour MP for Ilford North, said the Iranian regime is solely responsible for Ms Zaghari-Ratcliffe’s detainment.
But he added: “With the ill-judged and inaccurate remarks he made to the Foreign Affairs Committee last week, the only person who did the bidding of the Iranian regime was the Foreign Secretary.”
Anneliese Dodds, Labour MP for Oxford East, said: “We need now six words from the Foreign Secretary, ‘I am sorry I made a mistake’. Please give us those six words now.”
Mr Johnson replied: “I think the mistake, the error, the fault lies with the Iranian authorities. It is to them that she should be directing her anger.”
Conservative former minister Anna Soubry, writing on Twitter, said: “The lack of contrition is as shameful as the original error.
“Boris Johnson doesn’t understand magnitude of the job & responsibility he holds.”