Mr Mundell said he has reservations about the Prime Minister’s draft Brexit deal but other alternatives were “even more unpalatable”.
He told a gathering of Scottish Conservatives in Falkirk he would continue to press for reassurances the UK would leave the EU Common Fisheries Policy in December 2020.
Mr Mundell said: “She is tackling an issue of epic proportions, on which she can never please everyone, and she is doing her very best to find a way through.
“Let me be quite clear, if it comes to a confidence vote, she will have mine.”
The Scottish Secretary said Mrs May had this week demonstrated her strength of character, sense of public duty and “sheer decency in the face of an unprecedented onslaught by detractors and the media”.
Mr Mundell, meanwhile, accused Scotland’s First Minister of using Northern Ireland’s troubled past in attempts to increase support for Scottish independence.
Nicola Sturgeon has voiced concern the so-called backstop proposals for the Irish border would give Northern Ireland a competitive advantage by offering businesses easier access to the single market.
Mr Mundell said: “This week she reached a new low with her crass demands for Scotland to be equated with Northern Ireland.
“It confirmed that nothing is off limits now for Nicola Sturgeon - the sensitivities and unique circumstances of Northern Ireland, its geography, history and culture can be cast aside, if there is some opportunistic political point to be made.”
He added: “Scotland’s circumstances are nothing like those of Northern Ireland.
“Even the SNP used to acknowledge that, speaking out to have the Belfast Agreement upheld, but now none of that seems to matter any more, compared to the opportunity for tomorrow’s headline, one more seed of division, one more step however achieved on the road to independence.
“We, and Scotland, need to be on notice - Scottish politics just got a whole lot uglier. There is nowhere Nicola Sturgeon won’t go.”