She hit out after Mr Gray apologised to MSPs on Thursday for using a Scottish Government car to attend matches involving Aberdeen FC, the team he supports.
While he stressed he also took part in meetings at the games, he conceded he had given the impression of “acting more as a fan and less as a minister”.
Mr Gray also told MSPs that at several games he was accompanied by a family member or guest, and that they travelled with him in his ministerial vehicle.
The First Minister has already dismissed Tory calls for an investigation into whether the Health Secretary breached the ministerial code – which sets out how ministers should behave.
Speaking on Thursday, Mr Swinney said that Mr Gray had addressed “all the issues that need to be considered” in his statement, and that he considered the matter “closed”.
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But Ms Hamilton insisted that the Health Secretary’s statement had “completely undermined any claims that going to Aberdeen games involved Government business”.
In her letter she told Mr Swinney: “You must launch a ministerial code investigation today to uncover the full facts of what happened.”
The Conservative said: “These meetings blatantly didn’t involve Government matters if Neil Gray was joined by family members, friends or SNP activists.”
Calling for more details to be made public, she said: “If officials were there, we need to know who and what notes they made.
“Neil Gray must publish all correspondence around these trips, so the public can see if he was invited to Aberdeen games or asked to go himself, and he must also produce the minutes of all meetings and details of all conversations that supposedly involved government business, if they exist.”
The pressure on Mr Gray comes after his predecessor Michael Matheson resigned after he attempted to use public cash to pay for £11,000 of data roaming charges on his Scottish Parliament iPad – charges which were run up after the device was used to watch football on a family holiday to Morocco.
Ms Hamilton insisted: “Our NHS and social care system needs serious leadership, not another SNP minister out to swindle taxpayers.”
She added: “Neil Gray doesn’t seem to get that the public are furious that he misused taxpayer-funded limos to watch the football.
“If John Swinney won’t launch an investigation into this scandal, it’s clear he doesn’t get it either.”
A Scottish Government spokesperson said: “Mr Gray has made a statement to Parliament on this issue. The question raised by Rachael Hamilton was answered by the First Minister yesterday.”
