He was speaking in an emergency debate at the European Parliament in the wake of last week's EU referendum.
The UK voted by 52% to 48% to leave the union after 43 years of membership.
Mr Smith used his speech in Brussels to urge the EU to respect the will of Scotland after its electors voted by 62% to 38% to remain in the EU.
He won applause form the chamber when he said he was "proudly Scottish and proudly European".
Mr Smith went on to say: "I want my country to be internationalist, cooperative, ecological, fair, European.
"And the people of Scotland, along with the people of Northern Ireland and the people of London, and lots and lots of people in Wales and England also, voted to remain within our family of nations.
"I demand that that status and that esprit European be respected.
"But colleagues, there are a lot of things to be negotiated.  We will need cool heads and warm hearts. But please, remember this: Scotland, did not let you down.
"Please, I beg you, chers collÃ¨gues, do not let Scotland down now."
After his short address he received a standing ovation. The leader of UKIP and pro-Leave backer Nigel Farage did not join in the applause.
MEP Mr Farage had earlier spoken to his fellow EU parliament colleagues.
He told them that "the little people" had rejected "big politics" in order "to get their country back".
Mr Farage believed the UK now offered "a beacon of hope" across Europe for other countries that wanted to leave the EU.
He said:  "When I came here 17 years ago and said I wanted to lead a campaign to get Britain to leave the European Union, you all laughed at me. Well, I have to say, you're not laughing now are you?"
SNP MEP Alyn Smith received a standing ovation from his European colleagues after he begged them not to "let Scotland down".