Martial was shown a straight red card after an altercation with Lamela during the first half of Sunday’s Premier League clash at Old Trafford.
While Solskjaer admits it is ‘inexcusable’ for Martial to raise his arm to Lamela, the Red Devils boss was furious with the Tottenham star’s reaction as he fell to the ground holding his face.
Tottenham were 2-1 up when Martial was sent off and Jose Mourinho’s side scored four more goals after Manchester United went down to ten men to inflict a painful defeat on Solskjaer’s team.
Asked about the red card, Solskjaer told Sky Sports: ‘There’s no excuse [for Martial] but the boy [Lamela] must be having an operation on his throat.
‘It’s a bad reaction from Anthony and there’s no excuse because we are still in the game by then.
‘But if that [Lamela] was one of my players I would absolutely hang him up to dry because you don’t go down like that. He should not react like that.’
Manchester United made a superb start to the fixture, with Bruno Fernandes scoring from the spot after Martial won his team a penalty.
Tottenham bounced back immediately through Tanguy Ndombele and then Son Heung-min gave Spurs the lead, before Martial was controversially sent off.
Captain Kane scored his second and Tottenham’s fifth of the afternoon from the spot after Paul Pogba bundled over Ben Davies late on.
Solskjaer added: ‘It is the worst day of my career as Manchester United manager and the worst day for all of them as United players.
‘It is not the worst day in Manchester United history, we have bounced back before.
‘To explain now is quite difficult. We have been beaten 6-1 at home by a very good side but we have contributed a lot to our own downfall.’
Tottenham boss Mourinho, meanwhile, said: ‘Very good, very impressive but not a surprise for me.
‘The result is a surprise, you don’t expect to come here and win 6-1. I could come here 20 more times and not win by the same scoreline.
‘It happened because the team is phenomenal and has great character and great confidence.
‘We were the best team for all of the first half, we had to respect the club. We had to respect the history of Manchester United. They never give up. But we managed to get the three points.’