The 2m 55s YouTube clip features his rendition of the Frank Sinatra classic I'm Gonna Sit Right Down and Write Myself A Letter.
Mr Prescott pledged to post it if Dr Chand's bid for a parliamentary debate about the Health Bill passed 50,000 signatures. It passed the 100,000 mark earlier this week.
Mr Prescott tweeted:
Dr Chand is hopeful that the debate can take place in early March to coincide with a combined protest against the Health Bill.
He said: 'I'm very excited. This could be dynamite. This is a non-political protest, my passion has always been for the NHS.'
Asked whether he thought sufficient opposition was mounting to topple the Health Bill, he said: 'I hope so. I wouldn't be doing it otherwise.'