Can I also say stable?

Yes, "stable" can be used of a lasting, permanent relationship.

I personally would say 'steady' girlfriend but would more likely say a 'stable relationship'. Perhaps this is because of the connotations of "stable" (when used to describe a person) also meaning "sane, sensible, not easily upset or disturbed" etc. Thus an "unstable relationship" would be one that is not firm and fixed and may fail but an "unstable girlfriend" could mean that the girlfriend was prone to psychiatric problems or sudden changes of mood etc.

The above is my opinion as it relates to my dialect of English. I'm confident that you could use "stable" in the above sentence instead of "steady".

Thank you, that was very good explanation.

>I'd also say "steady girlfriend."...

Thanks CK. I thought it might be an Aussie thing but good to know US has similar reservations about "stable girlfriend". As "relationship" is not a person, I'd feel much better about using "stable" with it rather than with a person.

Related:

[#3171358] Tom has a steady girlfriend. (CK) *audio*
[#1093479] Tom is going steady with Mary. (CK) *audio*
[#1961736] I thought you were going steady with Tom. (CK) *audio*
[#1028755] Tom has been going steady with Mary for six months. (CK) *audio*
[#1893691] We were going steady. (CK)
[#3428576] Tom doesn't have a steady girlfriend. (CK)
[#491886] Unfortunately she already has a steady boyfriend. (swagner)