I'm going to assume by 'Roman bathing culture' you mean the provision of bathing houses, Thermae, such as the many across the city of Rome e.g. The Baths of Caracalla which included many other functions besides bathing (including a library). Reportedly 1,600 seats were available for bathers. Brick stamps attest to restoration works carried out under Theodoric (AD 493-526), indicating that baths were still considered worthy of continued maintenance well after they were built, as grand public buildings. The extent of the culture of bathing as being tied to that of the Romans is demonstrated by the still grand thermae found in far-flung provinces such as Britannia (such as those found at Bath, UK).

As culture is such a spurious subject it is difficult to pin down its exact ebb and flow. However, I wold argue that one of the predominant reasons for decline in public provision of bath houses (leading to a consequential decline in their use) came towards the end of the empire when wealth was retained for private rather than public expenditure. The decline in bath houses being then finished off (for the time being in the city of Rome) by the neglect of the aqueducts that would have allowed for the vast water supply needed to maintain the baths in working order.

However, to say that bathing culture disappeared everywhere to the same extent would be to disregard the vastness and complexity of the Roman Empire and the differing regional cultures therein. For instance the 'Turkish Bath' that was re popularised in the West by the Victorians (stemming from their fascination with health, British interests in the East and the re-imagining of the British Empire as the successor to Rome) could be said to have a real cultural link to the baths that would have been recognized by the Romans, including hot and cold rooms and pools, communal areas and attendants. However, the 'culture' was not fully replicated and the Victoria baths are far more associated with a particular luxury, rather than something that everyone would engage in every day.

I am not that knowledgeable on bathing culture in other countries by comparison. But if you were interested in Turkish Baths in the present day it's worth looking at modern spas that operate using the same system. e.g. this example in the North of England.

There are also some more recent (and more light-hearted and humorous) takes on Roman bathing culture in comparison to our own such as Thermae Roma, a Manga graphic novel and Live-action film that compares modern Japanese bathing culture to Roman's through the eyes of a time-travelling architect.