ICE not starting automatically at start-up

I replaced the 12V battery yesterday with a new one, and the car (2010 Prius Model III w Navigation), will turn on and off, but the ICE is not turning on automatically upon start-up, as it usually does. I can get the ICE to start by stepping on the gas pedal while in park, although it seems to die within a few seconds if I remove my foot.

The original issue was that the car died completely yesterday after parking. The same issue had occurred 3 years ago, and a new battery resolved it, hence my inclination that replacing the battery will work again. Everything else seems to be functioning normally, although I am not a technician.

I have searched around, and there are many comments on trying to reset the computer, clearing error codes, etc, but I am unsure what steps to take. Usually, the gas engine (ICE) will run for 20-30 seconds after turning the car on, even while in park.

The READY light DOES come on when turning the car on, and the ICE appears to turning on once driving above a certain speed. Also, there is plenty of gas. Does anyone have any suggestions or thoughts?

 

70k miles. The Dash went completely dark when the issue first occurred. MPG's are ~ 45 normally. The battery icon seems to be acting normal, is there anything to specifically look for?

In response to DonDNH- yes, I have waited a good 60 seconds to see if the ICE will start.

 

Just wanted to close the loop on this thread, for anyone who encounters this in the future.

For some unknown the reason, the ICE began its normal behavior 1-2 days after the 12V battery replacement. When starting the car (while parked), the ICE will turn on typically around 5-15 seconds after turning the car on. I wonder if the ICE turns on based on the overall temperature (suggested earlier) or certain car settings (e.g. if A/C is turned on). But, alas, the car is back to normal. Now, I hope I will get a good life out of this 12V battery. Thanks for everyone who chimed in.