Yes, 'I remember going swimming when I was in Ibiza' is fine. There isn't a problem with having two 'ing' forms one after the other.
The activity is 'go swimming' ( the same as 'go cycling', 'go skiing', 'go sailing' and dozens of other activities). The word 'swimming' never changes. Only the verb 'go' changes. You need to have the gerund of 'go' following 'remember', so the correct sentence is 'I remember going swimming'.