Anas Sarwar said it was important to take “political parties” and “personalities” out of the equation and focus on the code of conduct governing the first minister and members of her cabinet.
In an interview with Sky News, Mr Sarwar said that if a minister from another party had been found to have broken the ministerial code, Ms Sturgeon would call on them to resign.
“I think Nicola Sturgeon herself would say if an opposition politician was in government, and they had breached the ministerial code, then they would be expected to resign,” he said.
“Let’s take the party politics and the personalities out of it, it’s a point of principle and respecting the office of first minister.”
Ms Sturgeon said she first learned of the complaints against Mr Salmond in a meeting with him at her home in early April 2018, but it later emerged she had been told by his former chief of staff in her Holyrood office a few days prior, a fact she claims to have forgotten.
She referred herself for investigation by James Hamilton QC, an independent adviser on the ministerial code.
Mr Sarwar also said the controversy surrounding the Holyrood inquiry into the Scottish Government’s handling of the complaints against Mr Salmond demonstrated the need for reform.
He said that there should be a separation of the Lord Advocate’s existing roles as the chief law office to the Scottish Government and the head of the nation’s independent public prosecution service.
Mr Sarwar also used his first broadcast interview as Scottish Labour’s leader to claim that Ms Sturgeon would not support touting a referendum on independence were it not for the internal struggles in her party.
“The idea that we come through [the Covid-19 pandemic] and straight into a divisive referendum campaign, I just don’t think it’s the right thing to do,” he added. “Instead, I think it’s right that we focus on rebuilding our country.
“I actually don’t think even Nicola Sturgeon would be advocating a referendum right now, but I think she’s more focused on healing the wounds in her political party than she is about healing the wounds in the country.”
Meanwhile, Douglas Ross, leader of the Scottish Conservatives said “secrecy and sleaze” was threatening to consume the SNP leadership.
Writing in the Mail on Sunday, he said: “More than 50 requests from the Holyrood investigating committee and through freedom of information laws have been rejected. Witnesses have been shut down. Officials have been coached.”
Mr Ross said that Ms Sturgeon had misled parliament and broken the ministerial code, and argued that her position was untenable if her administration “continues to obstruct crucial evidence.”
Henry McLeish, the former first minister, said that if Ms Sturgeon had flouted the code, she must decide what course of action “would best serve the public interest.”
"Any individual in this position, in any party, must be able to recognise when a situation has become so damaging that allowing it to continue is not good for government or politics,” he added.