SATSQ: A Constitutional amendment is necessary to establish a right when the Courts do not recognize that right.
An Amendment is also useful to avoid a right being lost in future rulings, or to raise the bar on what is considered a compelling reason to abridge that right. Remember, DOMA was overturned not just because the Courts recognized freedom to marry as a right, but because the Federal government was unable to offer a compelling reason why it was in the public interest to remove that right from same-sex couples.
Consequently, a Constitutional amendment may well have been necessary to end slavery, because even if the Courts were willing to recognize both slaves as citizens deserving of rights and "not being a slave" as an unenumerated right in 1865 (hint: they weren't), the Courts almost certainly would have been willing to hear arguments that slavery was in the public interest, even if it was an abridgement of the rights of slaves. They are certainly willing to allow de facto slavery of non-citizens and of prison inmates to continue.
In any case, demanding that we refuse to recognize a right until it is enumerated in the Constitution or its Amendments, when the Constitution itself both declares the existence of unenumerated rights and demands they be recognized is just fucking asinine.