One of C++'s goals is to be a superset of C, and it's difficult for it to do so if it cannot support header files. And, by extension, if you wish to excise header files you may as well consider excising CPP (the pre-processor, not plus-plus) altogether; both C# and Java do not specify macro pre-processors with their standards (but it should be noted in some cases they can be and even are used even with these languages).
As C++ is designed right now, you need prototypes -- just as in C -- to statically check any compiled code that references external functions and classes. Without header files, you would have to type out these class definitions and function declarations prior to using them. For C++ not to use header files, you'd have to add a feature in the language that would support something like Java's import keyword. That'd be a major addition, and change; to answer your question of if it'd be practical: I don't think so--not at all.
