While diagnosing a problem like this is difficult without seeing the car, it sure sounds like you have a problem with either the starter solenoid or the ignition switch. The starter solenoid is an electromagnet used to start the engine while an ignition switch is the control system to a vehicle that activates its electrical systems. When these parts wear out, you won’t hear any clicking noise like you would if it was just an issue with the battery.
In either case, you will need to take your Acura TL to a mechanic to properly assess what’s wrong with it. Unfortunately, it’s not exactly a cheap fix if something is wrong with either the ignition switch or the starter solenoid. The cost to replace the starter is between $550 and $750 while the ignition switch could cost as much as $150 to $200.