Q: Push button won't start car, new battery

This may be related to a couple of different possibilities. You may have a faulty starter relay or possibly a bad ignition starter switch. The starter relay directs power from the battery to the starter solenoid which then engages the starter when you turn the key. When this is not working properly, you may notice intermittent starting problems including no response when pressing the start button. Generally, when this does fail, it does so very randomly and usually without warning other than intermittent starting. The power from the battery will still power other accessories such as radio, horn, etc., but you may not have any power to the starter. Your ignition switch has several different circuits that it goes through when starting the vehicle, any of which may fail causing the no start response you describe when pressing the button. I would recommend having a professional from YourMechanic come to your location to diagnose and inspect your vehicle.