The enumeration in the Constitution of certain rights shall not be construed to deny or disparage others retained by the people.The "right to keep and bear arms" is one of those enumerated (for those too stupid amongst you--it's enumerated in the Second Amendment).
If one wants to go down the road where gun rights are protected by the Ninth Amendment, one winds up running into this quote (thank you, Wikipedia) from Griswold v. Connecticut, 381 U.S. 479 (1965):
[T]he Framers did not intend that the first eight amendments be construed to exhaust the basic and fundamental rights.... I do not mean to imply that the .... Ninth Amendment constitutes an independent source of rights protected from infringement by either the States or the Federal Government....While the Ninth Amendment - and indeed the entire Bill of Rights - originally concerned restrictions upon federal power, the subsequently enacted Fourteenth Amendment prohibits the States as well from abridging fundamental personal liberties. And, the Ninth Amendment, in indicating that not all such liberties are specifically mentioned in the first eight amendments, is surely relevant in showing the existence of other fundamental personal rights, now protected from state, as well as federal, infringement.The problem, vis-a-vis Ninth Amendment jurisprudence, is that people have tried it too often in the past (as compared to Second Amendment jurisprudence) and been smacked down by the Establishment. The smackdowns have taken the form of decisions relegating the Ninth Amendment to being a rule of construction, rather than an affirmative rule which says "if you can think of a(n unenumerated) right, the Ninth Amendment protects it". This history has resulted in a welter of precedent which will as a practical matter make establishing any protectible rights under the Ninth Amendment likely a wasted effort and failure. I mean, if Douglas said "no Ninth Amendment rights" in Doe v. Bolton, 410 U.S. 179 (1973), as he did and as Ginsburg (who was one of the attorneys in the mix there) surely knows, who are you going to get to say otherwise.
Of course, the Ninth Amendment is construed to protect abortion rights (and birth control), which is why this argument is unpopular. In fact, any reference to the Ninth Amendment automatically drags in all the reproductive rights cases - Palko v. Connecticut (302 U.S. 319 ), Griswold, Roe, you name it - and the sexual conduct cases - Lawrence v. Texas (inter alia). You can also add proper medical care is something which is a benefit to society whereas firearms ownership has its detriments.
Additionally, if you are going to argue armed self-defence is covered under the Ninth Amendment, you have to get over the hurdle that the doctrine of self-defence, as understood at the time of the Constitution, stated that only the force necessary to counter the threat was appropriate. In fact, carrying a weapon could be construed as you were looking for trouble!
So, Abortion is OK even though "isn't mentioned in the Constitution", but Ninth Amendment case law supports it.
"Gun rights" are not covered by the Ninth Amendment since it is enumerated in the Second Amendment.