Richard Simpson also wanted licensed shops in problem areas to mark cans and bottles so they could be traced.
He told MSPs there was a problem with "proxy" purchasing for teenagers.
However, ministers argued that the issue could be better dealt with through the government's alcohol misuse strategy.
When the members bill was scrutinised at committee stage, BMA Scotland expressed concerns about some aspects of it but it supported the section aimed at banning alcohol advertising around schools.
Figures analysed specially for the Scottish government in 2009 estimated that Scotland had the eighth highest level of alcohol consumption in the world.
The World Health Organisation has linked alcohol to more than 60 types of disease, disability and injury.
MSPs have voted down a members bill which would have resulted in a ban on alcohol advertising within 200 metres (656ft) of schools.