It is possible the battery in your remote is dying...that signals a fail to start and a Check Engine Light for low remote battery voltage
Check the battery and the connections to the engine.
It sounds like a bad cable connection at the battery/starter solenoid/ or the starter solenoid is bad. Check these areas. It could still be the battery. Check the voltage at the battery then try to start the engine. If the voltage drops below about 10 or 11 volts, the battery does not have enough of a charge. Try a battery charger for a while and see if you can get enough of a charge to start the engine. If the voltage is up when you try to start the engine, check the battery post connections. See if you have voltage THERE when you try to start it. Sometimes a corroded battery terminal will act the way you described. If that isn't it, check the connections to the starter. If you have good voltage to the starter when you try to start it, the starter is probably bad.
Check your charge in your battery. Just because all things turn on does not mean your battery has enough power to start the engine. Charge your battery or get a jump start.
Probably a dead battery but you may wish to check your battery terminals for corrosion.
bad battery
You have enough battery power to power the radio, but not enough to turn over your engine. CHeck battery.
You don't want to do that. If your cars check engine light is on, you need to get it examined by a mechanic. Also, if you disconnect any battery terminal, your engine will not start or power any electronics while it is running.
Do you mean it won't start or it won't turn the engine over? Possibly poor contacts, check battery cable, also ground cables. If you have a sidepost battery, Remove the battery cables and check the center bolt that screws into the battery. These have a history of corroding and are easy to replace.
Was it running before the old battery went south? Did you have a check engine light before you replace the battery?
Check the engine, fuel, battery and if everything is ok with them and it still won't start it is best to get it looked at by a mechanic. Hope this has helped!
check battery, terminals, alternator