I'd have more of a problem with him trying to convince you he 'doesn't understand' (yes, he f*cking does) and being 'furious' with you. Does he have form for being a gas lighting piece of shit?
That reaction alone (even without her reaction) tells you all you need to know about his character. A decent partner and person would have felt horrible that this was making you feel insecure and want to do all he could to reassure you of his love.
Yes, men are aloud female friends. But he just met this lady and is getting all hot and sweaty with her and texting her. It is a little inappropriate for a married man and at the very least, he should understand your insecurities.
Just incase it is relevant - it is common for narcissists and the like to act like thy 'just don't get it' so that you focus on explaining why you are hurt in various different ways, hoping for them to understand. They DO understand! They just don't care!
Acting angry and things like calling you crazy or oversensitive are common behaviour from those sorts too. All to get you to look inwards and doubt yourself.
'Narcissistic triangulation' is something they may also do. Commonly it is when two people (eg you and the new gym buddie) are pitted against each other. Usually it starts by them name dropping this person and making you feel like they have a new shiny toy... and you are yesterdays news. But if you confront them about this feeling, they tell you you are wrong to feel that way. But instead of providing comfort, they act hurt/angry/accusatory or turn it round on you .
You have a right to feel the way you feel. Don't let him convince you otherwise.