Not exactly. Here's SCOTUS:

We find no merit in the remainder of appellant's contentions, including that of "involuntary servitude." As we have seen, 32 States prohibit racial discrimination in public accommodations. These laws but codify the common law innkeeper rule, which long predated the Thirteenth Amendment. It is difficult to believe that the Amendment was intended to abrogate this principle. Indeed, the opinion of the Court in the Civil Rights Cases is to the contrary as we have seen, it having noted with approval the laws of "all the States" prohibiting discrimination. We could not say that the requirements of the Act in this regard are in any way "akin to African slavery." Butler v. Perry,240 U. S. 328, 240 U. S. 332 (1916).

Basically, it said that when the Thirteenth Amendment was ratified, there were already laws like this on the books for innkeepers. Innkeepers are not providing a personal service, they are licensing lodging. Sexual services are absolutely personal services, and are easily distinguished from innkeeper obligations (that have long been defined under both the common law and statutory law, both before and after the Thirteenth Amendment).

Personal services are special under the Thirteenth Amendment. They are the reason why people cannot enforce employment contracts through specific performance on the employee (or any type of injunctive relief that requires a particular person to do a particular thing). It's why a doctor can say to a bankrupt person who has filed for bankruptcy, "I won't provide medical services to you until you pay your outstanding medical bills," but a landlord cannot say, "I won't continue leasing this property to you until you pay me my back rent."

And forced sexual intercourse starts looking a lot like something that is "akin to African slavery," to borrow a phrase from Justice O'Connor in United States v. Kozminski, 487 U.S. 931, 942 (1988). So yes, anything that looks like a personal service might raise a colorable Thirteenth Amendment argument, but specific personal services like sexual intercourse would almost definitely fall under the Thirteenth Amendment's protections.

It's not a clear cut argument, but it's an argument that needs to be addressed, and can't just be dismissed with a wave of the hand and saying "well Heart of Atlanta rejected that argument."