But he said he felt it hard to sympathise with him despite the American's confession that he cheated by doping throughout his career.
Armstrong, who is being sued by Walsh's newspaper, the Sunday Times, for more than £1 million over a libel payment made to him in 2006, issued the apology during the second part of his interview with Oprah Winfrey on Friday.
However, while Walsh said he accepted the apology in comments on Twitter, he expressed concerns about Armstrong's sincerity.
"Watching part two of Armstrong interview, he admits to feeling shamed and humbled. But why is it so difficult to empathise with his situation?" Walsh wrote.
"Oprah pressured him, the apology was, I thought, hesitantly promised. I didn't ask for it, or expect it, but, yes, if its offered, I accept."
The Sunday Times had earlier in the week taken out an advertisement in the Chicago Tribune with 10 questions they wanted Winfrey to ask Armstrong.
The English newspaper picked the Chicago Tribune as it is in the city where Winfrey hosts her show on her OWN cable television network.
Aside from the Sunday Times lawsuit, there is also a potential threat to Armstrong's liberty, stemming from the fallen icon's role in the US Postal Service team, where he spent his most successful years in the saddle.
Having been paid by the government, the former team leader could face criminal charges for making fraudulent statements to his bosses.
He could also be accused of perjury over disclosures made under oath to a US federal jury in 2005. If convicted, each false statement could lead to five years in jail.
Armstrong always maintained that he did not use banned substances during his stellar career, but the Sunday Times was one of the few publications to openly query his innocence.
The newspaper had to pay Armstrong £300 000 to settle a libel case after previously suggesting that he may have cheated.
The weekly paper is reportedly demanding the return of the original settlement payment, along with interest and legal costs.
Meanwhile, another of Armstrong's accusers says she is prepared to meet him now that he has come clean.
Emma O'Reilly worked as an assistant for US Postal during Armstrong's time in the team and was branded a "whore" by the Texan for contributing to a 2004 book that sought to expose his cheating.
In his interview with Winfrey, Armstrong admitted that O'Reilly's claims against him had all been correct and the Irishwoman said it was a weight off her shoulders.
"Lance responded to my accusations that he was a drug cheat by branding me a prostitute and an alcoholic," she told the Daily Mail.
"(In the interview), he said I was one of the people who had been 'run over' and that he owed me an apology, but more crucially he confirmed that my story about a back-dated prescription was true.
"There was this tightness in my chest that was suddenly released. I don't think I'd even realised, in all these years, that it was there. Only when it left me.
"He said he had reached out and tried to contact me. I want to speak to him but I'd rather do it face to face. I'd like to explain why I told my story in the first place." – Sapa-AFP.