What is Class F airspace, and why is it not used in the US?
ICAO Class F airspace is a bit of an odd duck (and the US FAA is apparently not the only agency that thinks so - from a quick check on Wikipedia it seems more jurisdictions ignore class F than implement it. They only mention Class F as being in use in Germany and the UK).

From a functional/regulatory standpoint Class F is a sort of hybrid between "Class E" controlled airspace and "Class G" (fully) uncontrolled airspace:
Class F airspace is designated as "uncontrolled", but you can get ATC clearances in Class F airspace just like in controlled airspace. The catch is they're "advisory only" (so you don't have to comply with them, and all the weight is on the pilot's shoulders ). Similarly ATC will provide separation services to IFR flights in Class F airspace, but they do so "where possible", which means they might tell you they can't provide that service and you're on your own with see-and-avoid.  

The general use case for Class F airspace seems to be allowing IFR flights to operate in "uncontrolled airspace".



As a US pilot (and thus having no experience with Class F Airspace), it sounds like the services provided in Class F airspace are effectively the equivalent of VFR Traffic Advisories ("Flight Following"): Controllers can give you advisories and make suggestions, but all responsibility is ultimately still on the pilot's shoulders, and you can politely tell the controller to get stuffed if you don't like their suggestions (without needing to declare an emergency to vest yourself with that authority).

In the US we already have the Flight Following system and culture set up for VFR folks (and generally speaking it's available nationwide, irrespective of airspace class, as long as the controller can get you on radar and isn't overworked). We also have a pretty robust Class E airspace nationwide.
Because of this the FAA probably sees no need to designate specific "Class F" regions where IFR flights can get the equivalent of "Flight Following": If you're VFR you can usually get flight following if you want it, and if you're IFR you should have no trouble staying in controlled airspace (Class A, B, C, D, or E) for your entire route of flight.