The Metropolitan Police has told Lord Bramall it regrets the “distress” caused to him during its inquiry into historical child abuse allegations.
However Assistant Commissioner Patricia Gallan stopped short of apologising in a lengthy statement, saying the force had a duty to investigate.
Lord Bramall, the former head of the Army, was interviewed last April.
The peer, aged 92, was told last week in a letter from the police that he would face no further action.
Lord Bramall, a Normandy veteran who retired from the House of Lords in 2013, was never arrested and always denied the allegations.
After being cleared, there wasn’t “one grain of truth” in the allegations, made against him by a man in his 40s, and accused the police of not behaving very well.
Friends said Lord Bramall’s late wife, who had Alzheimer’s, was at home when police officers “barged” into the house.
In the statement, Asst Commissioner Gallan said she fully recognised how unpleasant it would be to be investigated by the police over historical abuse allegations.
“For a person to have their innocence publicly called into question must be appalling,” she added.
She said she was taking the unusual step of explaining the dilemmas of policing after the possibility of an apology was raised.
“We have many serious allegations referred to us every year that we have a duty to investigate,” she wrote.
“It is, of course, a principle of British justice that everyone is equal before the law so that duty must apply equally to all, irrespective of their status or social standing.
“The fact that after a full and impartial investigation the evidence did not support charges being laid does not suggest that an allegation should not have been investigated.”
She added that the inquiry into the allegations continued until all lines of enquiry had been examined.
“The Metropolitan Police accepts absolutely that we should apologise when we get things wrong, and we have not shrunk from doing so.
“However, if we were to apologise whenever we investigated allegations that did not lead to a charge, we believe this would have a harmful impact on the judgements made by officers and on the confidence of the public.”
Ms Gallan said she would meet Lord Bramall to explain the force’s conduct at the end of Operation Midland, a wider police investigation into historical abuse claims.
Lord Bramall served during the D-Day landings during World War Two and commanded UK land forces between 1976 and 1978.
He became chief of the general staff – the professional head of the Army – in 1979, and in 1982 he oversaw the Falklands campaign.
Later that year he became chief of the defence staff, the most senior officer commanding the UK’s armed forces.