Can an aircraft operate IFR without ground-based navigation?
The AIM contains the information required to operate under IFR using GPS1.

To summarize the requirements,  WAAS (but not "traditional" GPS) may be used stand-alone without an alternate means of navigation provided:


Installation requirements:

The WAAS navigation equipment is approved in accordance with TSOC145 or TSOC146
The installation is done in accordance with AC 20-138

Operational Requirements:

The WAAS AFM procedures are followed
It is not be used by a commercial operator

Preflight Requirements:

The GPS database is current
The GPS database provider does not have any notices prohibiting the use of any fix or procedure that you are going to use
GPS NOTAMs/Aeronautical Information is reviewed prior to each flight 
A RAIM prediction for the time and route is done that shows no loss of RAIM
The flight plan navigation equipment is listed as RNAV



Note that commercial operators are required to have redundant systems or ground based backups in order to use it, and while not required for you, it's a pretty good idea.



1 More specifically, see:  

AIM Chapter 1, Section 1 - Navigation Aids  


1-1-18. Global Positioning System (GPS)

d. General Requirements
e. Use of GPS for IFR Oceanic, Domestic En Route, Terminal Area, and Approach 
Operations
f. General Database Requirements

1-1-19. Wide Area Augmentation System (WAAS)

c. General Requirements