He said his support came amid an “unprecedented onslaught” by her critics.
Mr Mundell told the local gathering he has reservations about Mrs May’s draft Brexit deal, but other alternatives were “even more unpalatable”.
He said 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.
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“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 that 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.