The apology was issued as SNP MP Patricia Gibson is understood to have been cleared in a separate case of sexual misconduct after an appeal.
Mr Blackford pledged to launch an external review of support available to staff, after SNP MP Joanna Cherry admitted the party had “significant problems” in how it handles complaints.
Mr Grady, 42, was found by an independent panel to have touched and stroked the neck, hair and back of a colleague 17 years his junior at a social event in 2016.
However, the SNP’s handling of the complaint against Mr Grady attracted criticism, most notably from the victim himself, who said his life had been made a “living hell” and threatened legal action against the party.
The row has been inflamed further, with audio leaked of a meeting of the SNP group at Westminster, in which Mr Blackford could be heard saying he was “very much looking forward to welcoming Patrick back into the group next week”, and encouraging fellow MPs to offer “as much support as possible”.
Following the leak, the SNP threatened its own MPs with criminal action.
In a statement issued on Tuesday night, Mr Blackford said: “As SNP Westminster leader, I have a duty of care to all of our staff. That is why I deeply regret that a member of staff was subject to inappropriate behaviour.
“It was completely unacceptable and should never have happened. I am sorry that it did.
"Staff must have full confidence that the group takes complaints seriously. In this case, the Independent Complaints and Grievance Scheme investigated what happened. We respect and accept that independent process.
"More than that, however, staff have a right to feel fully supported when a complaint is made. I regret that the complainant does not feel that this is the case.”
Addressing the audio leak, Mr Blackford added: “The way that this situation has played out publicly over the last few days, including recordings from the parliamentary group, has caused distress to the complainant amongst others and I am sorry that is the case.
"We will consider all lessons that must be learned to make sure staff have full confidence they will receive the support they need. As such, I am initiating an external review of support available to staff, to sit alongside the independent advice service and independent complaints process.
"Raising complaints of this nature is never easy, and I am determined that staff have the support they need."
In a statement in the Commons last week, Mr Grady said: “I am profoundly sorry for my behaviour and I deeply regret my actions and the consequences.”
In a separate case, Ms Gibson had been facing claims she made inappropriate comments to an SNP staffer in a Westminster bar in January 2020.
She has always strongly denied the allegations.
The allegations were probed by the House of Commons and initially upheld, but it is understood they have now been thrown out on appeal.
The decision by an independent expert panel will be published on Thursday.