Prof Hillis, who was the Scottish Football Association's medical advisor and Emeritus Professor of Cardiovascular and Exercise Medicine at Glasgow University, was involved in 228 full internationals.
He began working with the SFA in 1976 as Scotland Under-21s doctor and was promoted to the top team in 1982. He was club doctor with Clydebank for 27 years and also enjoyed a spell at Rangers.
Scotland manager Gordon Strachan said: "He was great company and was hugely respected in his job. There's nothing more you can ask of your team doctor but 'the Prof' always lit up the room with his personality.
"As a player you loved having him around and as a manager you could always rely on his advice."
Prof Hillis was vice-chairman of the UEFA medical committee and medical advisor to FIFA and was awarded the OBE in 2009.
SFA president Campbell Ogilvie added: "I have had the pleasure of knowing Stewart for almost 40 years and consider him a man of great knowledge, experience and wisdom.
"He was a medic of the highest standards and we are all fortunate that he brought those standards to the Scottish FA during his time as international team doctor and medical director and advisor.
"As well as his professional capabilities, he was also terrific company and I know that his personality, sense of humour and repartee were as essential to many Scotland national teams under many national coaches as his medical expertise.
"My thoughts go to his family: his wife, Anne, his sons Andrew, Ally, Iain and daughter Sara. He will be sadly missed by his many friends at the Scottish FA and throughout the football world."