Is isolation required for MOSfet gate driver's and microprocessor's power supply?
In the comments you say your concern is that noise could couple to the MCU through the driver's input pins.

According to the datasheet,


  The input stages of the UCC27201A incorporate an open-drain
  configuration to provide the lower input thresholds. The input
  impedance is 200 kΩ nominal and input capacitance is approximately 4
  pF. The 200 kΩ is a pulldown resistance to VSS (ground).


Absolute maximum voltage on inputs LI and HI is 20V, which indicates that they do not have protection diodes connected to Vcc. Therefore the only way that noise from the outputs can get back into the inputs is in reverse through the output driver, buffer, and input FET. With a Gate capacitance of just 4pF the noise will be greatly attenuated at the input pin, if it even gets that far. 

One reason to use a driver IC is that it provides excellent protection against spikes coming back through the switching FETs. If enough noise does get back through the driver to affect the MCU then you probably have a more serious problem, such as the driver blowing up.