David Mundell was the only politician who met the US president when he landed at Glasgow Prestwick Airport for his weekend in Scotland in mid-July.
Trump did not issue an invitation to the First Minister or any other Scottish ministers to meet him while he was in the country.
Mundell suggested that an invitation from Trump would have left the SNP leader with a difficult decision to make.
He told the Sunday Mail: “I understand Nicola Sturgeon wasn’t asked to meet him. It’s my view that she should have been asked and then she would have had to make clear whether she was going to do that or not.”
He added: “I felt they let her off the hook - she should have been part of the welcome.”
Sturgeon made it clear she would not refuse a meeting with the president, but said that she would be uneasy in his company.
Mundell also desrcribed the “surreal” moment he welcomed Trump to Scotland.
He said: “I was on the tarmac when Air Force One came in to land. Trump came down, he put his hand on my shoulder and said, ‘Hey, how are you?’
“It was a much easier conversation than it might have been. He said he’d been in Scotland many times before but this was his first visit as president. He talked about how much he loved Scotland, how his mother was Scottish and how happy he was to be here. Then it was all about golf.”
While refusing to support Trump’s policies, the Scottish Secretary felt it was important to give him a formal welcome.
“He has a charisma, not everyone thinks it is positive but he’s not a bland person and that has projected him to where he is,” Mr Mundell said.
“He’s a head of state and that’s why I met him. The US is one of our most important allies and I think you need to take that on board. I fully respect people’s right to protest but having strong links to America, from a business and political point of view, is important. You need to take those forward regardless of who is president. One thing I felt was that he has a genuine affection for Scotland.”
Trump’s controversial animosity towards Sturgeon was revealed ahead his visit to the UK.
A number of former advisers to Prime Minister Theresa May were quoted in an article about how difficult it is to keep conversations between the two leaders on track.
One gave his ‘hatred’ of Nicola Sturgeon, and her predecessor Alex Salmond (with whom Trump clashed over windfarm developments near his Aberdeenshire golf course) as an example.
The former staffer told the Huffington Post: “He totally hates Nicola Sturgeon. He spends lots of his time bitching about Sturgeon. He loathes Salmond too. But why spend so much time talking about Sturgeon in a phone call with Theresa May?”