One does not need to cover his body in a bathhouse.[2] The same applies to a river. (Nevertheless by a river one should get undressed and dressed as close as possible to the river in order not to be uncovered unnecessarily).[3] [From here it is learned that one should not leisure around even in a bathhouse when he is not dressed. Based on this some are particular to remove their under garments close to the Mikvah.[4]]

Covering the Erva: Upon descending to the river one does not need to cover his private part, and one who does so appears to be embarrassed [of his circumcision] and its considered as if he denies the covenant of circumcision made with Abraham. However when ascending [from the river], being that one is now facing towards people, one should bend [in a way that will prevent his private part from being seen], or should cover his private part with his hand in a way that he will not touch it. This likewise applies when one is bathing in a bathhouse together with other people. If it is possible to cover one’s Erva from the eyes of others using a cloth then this is the best option.

Summary:

It is permitted to be unclothed in a bathhouse or river. Nevertheless by a river one should get undressed as close as possible to the river. Upon ascending from the river/bath one should his cover his erva with his hand or a cloth.

Q&A

May one remain not fully clothed by a beach?[5]

One may remain in the water undressed although he is not to walk around the beach with his body uncovered.

Is one to cover his Erva when bathing in a Mikveh?

One is to cover himself when returning from the Mikveh back into the dressing room.[6] However when entering into the Mikvah from the changing room it is unclear from Admur as to whether one is to cover himself in this instance.[7] Practically the custom is to be lenient not to cover oneself at all in a Mikveh.[8] Although meticulous individuals are careful to cover themselves.[9]Alternatively they remove their undergarments only near the Mikveh.