1st to ask is the new thermostat the same specification as the old one?
as they do come in a few opening temperatures, if you look on the body of the thermostat there will usually be a number stamped on it like '78' or '82' this is the opening temperature in centrigrade
I'd also suggest ensuring that you've got all the air out of the system as this sort of episode can leave pockets of air trapped and prevent the coolant from circulating properly.
So check that the small hoses that run from the top of the engine and radiator that run back to the header / expansion tank are free flowing and not blocked
If these are OK then remove the tank pressure cap and run the engine from cold with the heater set to full hot
As the engine starts to warm, squeeze the large hoses running to the radiator to help displace any air - all the time keeping an eye on the level in the header tank, making sure it doesn't drop below the minimum and open any bleed screws built into the top cooling hoses - closing them off as soon as they start to gush water
Once the engine is up to temperature the tank should be quite full. Switch the engine off and leave it to cool
Then top up to the mark once its all cold and replace the cap
If all of this is OK it may be worth replacing the header tank cap as again you can lose coolant as steam if this is blowing off too early ( the cap should hold pressure of around 1Bar)
if its still got an issue after this and its still spitting water out then I'd agree its wise to test for head gasket
The other thing to try is a head gasket / block test kit as these take a coolant sample and analyse it for any signs of combustion gas - if present this would indicate the head gasket is on its way out