If I change the access modifier of the setter of a public property from private to public, is that causes any breaking change in the other assemblies that refer it?
Since the visibility of the property is increased now there should not be any problems with statically linked assemblies and static references -- those assemblies just treated your property as read-only until now and they continue doing so.
The only problem that might potentially arise here is if some assemblies use reflection to get information about your properties and perform some actions.
For example, if this is a property of a business object and you use data binding to show your data in some control, like GridView
or ListView
(independent from the actual framework), it might be that this property will be recognized as "updateable" now and the user might change some values you (and some other assemblies) considered immutable so far.