Ross Thomson, who won in the first ruling by a new independent expert panel and in an appeal last month, also called for Paul Sweeney to be investigated and suspended from Scottish Labour.
Mr Sweeney had claimed Mr Thomson had groped his backside and genitals in a House of Commons bar on October 30, 2018.
In a letter sent to Mr Sweeney, Mr Thomson’s solicitors Ledingham Chalmers also said the former Aberdeen South MP was considering defamation action as a result of the "substantial" damage to his political career and reputation.
The letter read: "My client's greatest desire is to be able to uncontrovertedly clear the stain on his character caused by these allegations.
"While it may be possible to put a monetary value on the damage ... a clear written apology from you ... would in his eyes be the most satisfactory compensation.
“Were he to receive such an apology, my client is prepared to draw a line under matters and take no further steps to pursue you for the losses arising from your false complaints."
Mr Sweeney, 32, who lost his seat in the 2019 general election, is now standing to be an MSP and is third on the Glasgow regional list.
But a separate letter from Thompsons Solicitors, addressed to Scottish Labour leader Anas Sarwar, called into question Mr Sweeney’s fitness for candidacy.
“It is of some surprise that Mr Sweeney has retained his membership or is being considered as a potential candidate for election,” it read.
"If the party has not made [any] investigation into this matter to date, please treat this letter as a formal complaint against Mr Sweeney and confirm that it will be investigated under [Scottish Labour’s] disciplinary rules.
"Pending that investigation concluding and given the seriousness of the false allegations made, it would also seem appropriate to suspend Mr Sweeney's membership at this time."
A Scottish Labour spokesman said: "We confirm receipt of this letter and will respond in due course. All complaints are handled in line with party procedures."
Mr Sweeney was approached for comment.