Scotland voted in favour of the UK staying in the EU last week by 62 per cent to 38 per cent - with all 32 council areas backing Remain.
However, because the UK as a whole has voted to Leave - this raises the prospect of Scotland being taken out of the EU against its will.
Gil Paterson said: “The independence supporters of Milngavie and Bearsden must now rally together after Nicola Sturgeon announced a second referendum is on the table.
“We pledged in our manifesto to support a second referendum on Scottish independence if we were dragged out of the EU against our will.
“I was up all night watching the coverage and the result was absolutely tragic for the whole of the United Kingdom. “However, I am so proud of my constituency and all other parts of Scotland for their landslide support of remaining in the EU.
“This irreversible decision by people mostly in England has changed the United Kingdom forever.
“I appeal to everyone in Clydebank and Milngavie, and especially those who voted no in the 2014 referendum, to rally around Scotland remaining inside the EU as an independent nation.
“The EU referendum has proven that Scotland can unite in one cause so let’s do it again for all the best reasons.”