No, there isn't.
The whole point of Revlon duties is that they trigger "in certain limited circumstances indicating that the 'sale' or 'break-up' of the company is inevitable" [1]. Outside those conditions, "the singular responsibility of the board" is not "to maximize immediate stockholder value by securing the highest price available."
> There is no such requirement for a private company
Are you thinking of minority rights? These vary based on whether a company is closely held or not [2], not whether it's public or private.
[1] https://en.wikipedia.org/wiki/Revlon%2C_Inc._v._MacAndrews_%....
[2] https://millerlawpc.com/rights-minority-shareholders-private...