I hurt my leg sounds perfectly fine in this situation.

It is perfectly correct. Both of the following sentences are correct, and mean different things. 1) "I hurt my leg" means "I injured my leg." Perhaps the injury is serious, perhaps not. 2) "My leg hurts" means "I am feeling pain in my leg." So, this sentence makes sense: "I hurt my leg, and it's hurting a lot." So does this one: "My leg hurts, and I don't know why, because I don't remember hurting it."

- I hurt my leg. - I hit/banged/bumped/smacked my leg on the corner of the tea table. - I rammed/knocked my leg into the corner of the tea table. - Ouch! ;)

You can say "I have hurt my leg" but it is more common to reduce this sentence to "I hurt my leg".

Did you mean 'could he have said'? What he said is perfectly fine, and natural, and he is perfectly entitled to say that, so 'should he have said' does not make sense. He could have said 'I bumped my leg'. He could have been more specific, and said 'I bruised my leg'.