He will take on the four-year role from 1 August this year after being selected by the Scottish Government's public appointments office.
Robinson, who will replace outgoing chairman Ian Kinniburgh, joins the health board at a time when it needs to cut costs by millions as it tackles high expenditure.
He said he was "really pleased" to be taking on the job, but stressed that it is a challenging time for public sector organisations.
"The experience that I have had in local government I think is very relevant in so far as that the council had to deal with very much of the same challenges that are facing the health board," Robinson said.
"Clearly one of those is the ageing population. People are already living longer, but we want to see people living healthy lives for longer and I think that’s a challenge for the health board, but also for the council and its community planning partners."
Robinson added that he would like to see the "kind of forward planning that was done in the council to look at what services are going to look like five years, ten years down the line, so that we can plan early to meet to those challenges."
He also said there’s "more opportunities" to be had from telehealth in Shetland if the Scottish Government sticks to its commitment of providing 100 per cent broadband coverage across the country by 2021.
An announcement on his appointment said Robinson "oversaw the implementation of the council's governance review and successfully led the council back to sustainability following many years of overspending.
"During this time, he was also responsible for the successful integration of health and social care services in the islands.
"A strong advocate of community empowerment, he personally led several participatory budgeting exercises throughout the islands and was a commissioner on Shetland's inequalities commission."
It added that "alongside his counterparts from Orkney and the Outer Hebrides, he co-founded the award-winning Our Islands Our Future campaign which culminated in the Islands Bill currently passing through the Scottish Parliament.
"Gary has been a member of the Chartered Institute for Sport and Recreation Management since its inception."
Robinson was first elected as a councillor in 2007 before he was appointed leader in 2012.
He failed to regain his place in the Shetland West ward in last year's council election.
The NHS Shetland chairperson role has a remuneration of £28,288 and the job is part-time with a commitment of three days per week.