The constitution does not say anything about 'separate but equal', for or against, directly. So the easy answer is that separate bathrooms are not a violation because you can't violate something that is not constitutionally required.
Of course that's sort of a cop-out, so let's answer the question you're trying to ask without being so overly literal. I'm sure you're referring to segregation and why segregation was ruled unconstitutional yet segregating people by sexes when they use the restroom is not.
Quick sidenote: I'm focusing on cisgender individuals for this question. One could argue that the situation is different when you consider transgender, or other types of genders, individuals. I'm ignoring that group for now to keep this answer from growing too long, and I don't think it is what the OP is interested in right now. Let's just say a separate question about how separate but equal concept applies to transgender individuals' use of restrooms could very well be asked and discussed
The short answer would be that the two sexes are actually being treated equally, whereas segregation was not leading to equality between the two groups that were segregated. To better explain what that means, let's first go over a quick history lesson.
History of Segregation
Segregation was ruled to violate the 14th amendment. Specifically the part offering equal protection. Here is the relevant section:
All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the state wherein they reside. No state shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any state deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.
You notice that although the 14th amendment was written specifically to protect the rights of former slaves, the wording does not limit itself to former slaves, the fourteenth amendment applies to everyone! (Though I find it ironic that the very next section offers all males the right to vote, but still excludes females from voting, luckily later amendments have fixed that.)
The most relevant part of the amendment is that last line, stating that all US citizens are offered 'equal protection of the laws'. This means that no law could be passed that offered one group of people (whites) a privileged to use some resources (such as riding the bus or drinking out of a water fountain) while denying another group (blacks/former slaves) that same privilege.
Originally the Supreme Court ruled that this did not prevent 'separate, but equal' treatment. So, for example, you could not forbid one group from drinking out of a water fountain, but you could pass a law saying both groups had the right to drink out of a water fountain, but provide different water fountains for each group. Since both groups gained the same privilege and were treated 'equally' the Supreme court argued this did not violate the equal protection under the law offered by the fourteenth amendment.
The problem was the 'but equal' part. Sure, the Supreme Court suggested that the two groups would be treated equally, but in practice the water fountain for the 'negro' people were always worse than those for whites, not cleaned as well, not maintained, etc. In general the items and locations provisioned for 'negro' use were uniformly worse than the ones provisioned for whites to use. Thus the Supreme court later ruled that the idea of treating the different races 'separate, but equal' was inherently flawed because in practice the separation was being used to treat one group ('negro' people) worse than the other and thus the law was not being applied equally amongst the two groups.
In fact the supreme court went a bit further, by stating that the very act of treating the two races of people separately in this case was harming one of them. Even if both sides had exactly as nice of a water fountain, and had just as desirable of bus seats etc, everyone would know that the reason that the two were separated was because one group was considered to be less than, and thus not worthy of being allowed to interact with, the other group. Telling a group of people they were considered less worthy by segregating them will cause that group to suffer psychological harm, feelings of inferiority or anger at the treatment etc. This was an unfair harm being inflicted on one group, but not the other, by the very act of segregation, and this meant that it was impossible to ever have truly separate, but equal, treatment of the races.
So why doesn't this apply to separation of the sexes?
So why wouldn't this apply to bathrooms? Mostly because the concept of separate, but equal, was not actually illegal by the fourteenth amendment. The problem was that in practice making a division based off of race always told one race they were inferior to another.
When dividing two groups based off of sex this isn't necessarily the case. First and foremost the two sets of bathrooms are generally treated truly equally, unlike what happened with segregation. Both bathrooms are cleaned as frequently. Both are the same size with the same degree of decoration etc. One isn't treated worse because the owners of the bathrooms consider one of the sexes to be somehow inferior to the other. (This is a slight lie, as I believe studies show female bathrooms are cleaned slightly more thoroughly, as women are more likely to complain about unclean restrooms then men are, but the difference is very minor and not due to some desire to slight one over the other.)
Second, and more important, there is no inherent harm in separating the two bathrooms. When you had a separate restroom for whites and blacks, you were effectively telling blacks they were inferior and not worthy of using white's restrooms. This is not the case with separating restrooms between sexes. I don't think either sex feels that the reason they have to use a different restroom than the other is because they are inferior (or superior) to the other sex. This is important, separating the sexes, in this situation, is not causing any inherit psychological harm by its very nature!
Going along with the last point the fact that there are valid reasons for separating the sexes, both because the majority of both sexes would prefer to be separated in the restrooms and because the manner of constructing the restrooms differs due to legitimate differences in the sexes (as males have space dedicated to urinals that females don't) helps to justify a reason for separating the sexes and thus further decreases the odds that anyone will view the separation as a statement against one sex or the other.
I would also point out this is not the only case that separate-but-equal policies exist. There are many situations where two groups are treated separately, but not with an intent to provide unequal harm to the other. As a random example whenever I vote the location I go to in order to vote is based off of my last name, if my neighbor has a last name that starts with a different letter he may be told to go to a different voting location. We're being separated into groups and told to use separate accommodations. However, since the accommodations are generally equal, and the separation is not done with an intent to mistreat or denigrate one group over the other, it's not a violation of the 14th amendment. There are many many cases where it makes sense to separate people into groups, usually for logistical reasons, that are not a violation of the 14th amendment for this reason.
When/if the separation of sexes by bathrooms is proven to cause a disproportionate harm to one group of people over the other then an argument of violating the 14th amendment could be made, but not until then. I will now promptly end this answer before anyone brings up transgender individuals by simply saying that some have made arguments that the transgender individuals are not getting 'equal protection under the law' due to how they are treated with regards of which restroom they may use. I'm sure a separate interesting question could be asked as to how the 14th amendment may be applied to transgender individuals and the law, and I don't have time to get into it right now ;)