STV, who screened the first debate between the two rival politicians last night, said 1.2 million had tuned in to watch the debate on television, with another 500,000 watching online.
Bobby Hain, director of channels at STV, said that gave it the highest audience for any political debate in Scotland for the last decade.
It attracted more than a third of viewers (36%) in Scotland, according to the STV, making it the best viewed programme at the time.
Voters north of the border will go to the polls on September 18 to decide if Scotland should remain in the UK or become an independent country.
The social media site Twitter recorded 186,000 tweets about the debate, peaking at more than 2,000 a minute.
The two-hour long clash was being broadcast on STV, on ITV Borders Scotland via Freeview and online at www.stv.tv.
But a surge in demand online at the start of the debate caused STV's livestream to crash.
Mr Hain said: "This landmark TV debate delivered ratings that make it the best watched political debate in Scotland for more than 10 years."
He added: "We are aware that, although the live online stream achieved over half a million viewers, due to extremely high demand some viewers had difficulties accessing the programme. We apologise to anyone who was not able to watch the debate in full. The programme is available on our catch-up service, STV Player."