Did you ever figure this one out?
That starting circuit is pretty simple. Based on a quick look at the wiring diagram there is a large wire that runs from the battery to the starter that is hot all the time. There is also a smaller brown wire that is connected to one of the solenoid terminals that is also hot all the time. There should also be a red wire on the Solenoid that is not hot except during starting. It would appear from the wiring diagram that when the key is turned to start the car the ignition switch should electrically connect the brown wire to the red wire, which engages the starter solenoid to make the starter crank. The only parts in that system are the starter motor, starter solenoid, ignition switch and and the wires themselves.
The only possibilities that I see for your problem are:
1.) Bad battery (you can still measure voltage but if the voltage is low but the starter won't engage and the car won't start),
2.) A broken wire between the starter solenoid and the ignition switch,
3.) Bad starter solenoid (being new does not exclude this),
4.) Bad starter (being new does not exclude this),
4.) A bad ignition switch.
To test: Jumper the red and brown wires on the starter solenoid (a screw driver will usually work for this if you are very careful, a remote starter push button is better) If it cranks it's a broken wire between the starter and the ignition switch or a bad ignition switch. Given Fiat's rather questionable practice of running everything electrical in the car through the ignition switch that is often the culprit.
You have probably already had this fixed but believe it or not people actually look at old posts to solve problems that they encounter. When there's no answer it's not real helpful.
