Does the product on USA market with inactive transmitter (2,24 GHz) need the FCC certification?
We are going to use Tx/Rx module, but the transmitting function will be software disabled.
Yes, I think you do need FCC certification anyway, the reasons for this are:
Even a receiver "transmits" as it will have some internal oscillator needed to receive the signals.
EMI restrictions, almost any electronic device emits some form of electromagnetic waves and this must be below a certain level as to not interfere with other devices. These interfering emissions not only originate from transmitters, also switched power supplies and high speed digital electronics can be culprits.
If you use a module which is already FCC certified and make sure the electronics you add also meet the FCC requirements, you should pass FCC certification.