Though state laws differ most states have a law requiring a front windshield to be in place in an automobile for safety reasons and for it to be made of safety glass. Most states in fact require a windshield and working windshield wipers for street legal automobiles.
For example Illinois law does not allow a person to drive a car with a windshield, side or rear window that is needs to be repaired or is defective in that it impairs the driver's view to the front, side or rear.
According to Section 56-5-5010 of the South Carolina Code of Laws, titled Safety glass in motor vehicles, no person shall sell any new motor vehicle nor shall any motor vehicle be registered unless such vehicle is equipped with safety glass wherever glass is used in doors, windows, and windshields.
New Jersey title 39 subsection 3-75 (39:3-75) discusses windshield and safety glass. Here it states that among other things that no person shall drive any motor vehicle manufactured on or after July 1, 1935 and registered in this State unless it is equipped with approved a windshield and safety glazing material.
The requirements for a windshield, windshield defrosting and defogging, windshield wiping and washing systems, and glazing materials are also contained in Federal Motor Vehicle Safety Laws. The National Highway Traffic Safety Administration (NHTSA) has a legislative mandate under Title 49 of the United States Code, Chapter 301, Motor Vehicle Safety, to issue Federal Motor Vehicle Safety Standards (FMVSS).
These Federal safety standards are regulations written in terms of minimum safety performance requirements for motor vehicles or items of motor vehicle equipment. These requirements are specified in such a manner "that the public is protected against unreasonable risk of crashes occurring as a result of the design, construction, or performance of motor vehicles and is also protected against unreasonable risk of death or injury in the event crashes do occur."
If your friend was already advised by law enforcement that he could not drive the vehicle until the windshield is replaced then he cannot do so, even with safety goggles. Goggles may be able to keep little bits of debris out of his eyes to help him see better it cannot keep rain, snow, hail, etc from impairing his vision since a windshield will have windshield wipers and his goggles will not. Also there could be large chunks of debris such as tire bits, wood, etc that fly off other vehicles or the roadway and that would normally bounce off your windshield but now would hit the driver or passenger.
Your friend can check with your state's Department of Motor Vehicles to find out the exact laws regarding windshield glass in your state and what the penalties might be if he was stopped driving without a replacement windshield in place.
Hopefully the DMV will advise your friend of other options he may have instead of driving his car without a windshield since this poses safety issues such as items flying into your car and possibly injuring the driver and any passengers.
Your friend should also keep in mind that without replacing the windshield his vehicle could be damaged further by the sun, rain, etc if he does not at least cover the front windshield opening before he gets it fixed.
If your friend has comprehensive coverage on his car he can make a claim with his auto insurer. Glass breakage falls under this physical damage coverage. Also since a psycho girlfriend, hopefully an ex-girlfriend now, did the damage he should be able to claim it as vandalism. If he does not have this type of car insurance coverage then he still should get his windshield repaired and then sue the at-fault person for the cost.
The idea of driving with goggles may sound okay since those with dune buggies, etc may do it, though in many states dune buggies are not street legal or are not allowed on high speed roadways such as the freeway, but it really is a safety issue. Cars go at much higher rates of speed then most off road vehicles that allow safety goggles and not front glass and are on the roadways with more debris and chances for something to come through the area where the windshield should be.