If your RC car used a Bluetooth radio to control it (I'm sure there are some out there), then obviously it is possible and does not require any modification, just the right app on your Bluetooth-supporting phone.
So I'm assuming your question is about the traditional sort of RC car with a dedicated transmitter/controller, and about the general reasons it probably won't work. There are several problems at different levels, but it all comes down to the two ends are too different.
Frequency. Radio transmissions are at different frequencies both as a fundamental fact of physics (you have to pick one) and to avoid interfering with each other (pick one that someone else isn't using).
There are many physical/electronic factors making any radio more or less specifically built for a given range of frequencies — the antenna, local oscillator, and filters (if any) are all built for that range and will operate poorly or not at all outside of it. Some radios are designed to work across a very wide frequency range ("general coverage" receivers, research/hobby SDRs, etc.) but as a result of this they have worse performance at any given narrower range. And for cell phones, the radio performance — whether you have usable service or not in a given location — is a key selling point.
Firmware and modulation. The radio chips in your phone are not completely general-purpose SDR radios — they are designed to "speak" a particular radio protocol (GSM, CDMA, Bluetooth, WiFi, etc.). The specific design allows them to be much more energy-efficient (longer battery life for your phone), but means that they (most likely) cannot be told to switch to the frequency your RC car expects to receive, or to send the format (modulation, protocol) of control signal it expects.
Software. This is the least fundamental of the issues, but still one of them. Even if your phone's radio hardware could be "convinced" to do what you're looking for, you would have to have some way for your application to take control of the radio hardware. Generally, a phone operating system does not have a provision for this, and if it did there would need to be a feature for it also in the driver provided by the radio chipset's maker.
There are also legal/regulatory issues with being able to program a radio to do something different. In most countries, devices containing radio transmitters which are sold to the general public are required to operate in specific expected ways (specific frequencies, maximum power limits, obeying various more specific rules) and be unable to be reconfigured readily to send arbitrary radio signals.