The Liverpool left-back will wear the armband against Belgium in the friendly match at Hampden Park on Friday which precedes the Nations League game against Albania the following Monday.
Robertson made his Scotland debut in a 1-0 win over Poland in 2014 and has scored two goals in 22 appearances.
His first goal for the national team came in a 3-1 friendly loss to England at Celtic Park in November 2014, with his second coming in a 3-0 win over Lithuania in 2018 World Cup qualifying.
Meanwhile, Swansea striker Oli McBurnie has withdrawn from the squad due to injury.
McLeish will bring in a replacement who has not yet been named.