Anas Sarwar said it is “her responsibility” and said decisions made so far will not resolve the issue.
He said she should apologise and “consider her own position”.
His comments follow the health board’s chairman Professor John Connell stepping down on Friday after Ms Robison called for him to quit when it emerged NHS Tayside had apparently used donations to fund new technology.
Mr Sarwar said: “She should immediately apologise to the families who donated money in good faith - and explain when the SNP Government will properly fund NHS Tayside.
“This scandal took place on her watch, in her own backyard. It is absolutely essential there is a full and swift investigation by OSCR, as well as a national probe to reassure the public this is not going on elsewhere.
“If she thinks simply sacking a chairman she herself appointed just two years ago will resolve this scandal, she is not just incompetent but negligent.
“SNP Health Secretary Shona Robison cannot wash her hands of the scandal at NHS Tayside. It is her responsibility and, if she had any sense of decency, should be considering her own position.”
Earlier this week it emerged that NHS Tayside took more than £2 million from its endowment fund - which is made up of donations from the public or bequests in wills - to cover general running costs, which could normally be funded from its core budget.
The health board, which was bailed out with a Scottish Government loan of £33.2 million in 2016-17, was reported to have used the endowment fund when “faced with a funding deficit” in 2013-14.
It was further claimed NHS Tayside had to temporarily suspend its constitution to allow this to happen, as the money was going to retrospectively fund projects already approved by the board.
NHS Greater Glasgow and Clyde chairman John Brown has been appointed as interim chairman of the Tayside health board.
The Liberal Democrats have also called for Ms Robison to apologise.