Im using TC1412 MOSFET drivers on an H-bridge for a motor, the datasheet says the drivers are protected against 5kV of ESD.
Is it necessary to use a flyback diode on the output?
Two quite different answers are possible because the question is ambiguous.
The TC1412 MOSFET gate driver IC is intended to drive an H bridge made of external MOSFETS BUT is a half H bridge in its own right so two could be used to drive a motor directly - hence the uncertainty.
" ... on an H bridge" may mean
Either is a possible and valid interpretation as the IC itself is half of an H bridge. ie Are you driving a motor directly with this IC, or using the IC to drive a FET H bridge to drive a motor? Those answers affect the answer to your question. So the two answers can be summarised as:
Use as a direct motor driver.
Assuming that you mean that you are using the TC1412 as a motor driver directly, which is a valid use but not what it is designed for, and assuming that you may use it at its rated current of up to 2A, then there is nothing in the datasheet that guarantees it will survive an inductive transient. The datasheet says that it will survive up to 500 mA injected into its output transistors. By (strong) implication, it may not survive currents higher than that. If you are using it at up to 500 mA you do not strictly "need" diodes or similar protection but they would be wise.
Note that what inductive protection is needed and what "topology" is needed depends on your application. If you are implementing a bidirectional full H-bridge motor drive with the motor being driven directly by the IC then you cannot just connect diodes across the motor as they would be forwards biased when the motor polarity reverses. in such cases you will still probably have to deal with inductive transients but must direct and dissipate them in some other way - eg with RC snubbers or catch diodes to supply or whatever. If you are driving a motor in one direction only then a "flywheel diode" across the motor will be OK.
MOSFET gate drive
If you are in fact using the IC to drive MOSFET gates and NOT a motor directly then you do not need formal flyback diode at the MOSFET gate. BUT you may be advised te use either-
Please add a data sheet reference when asking questions related to a specific IC - I've added one to your question.
They did not directly state that there was internal protection, there was a 4KV electrostatic discharge protection but weather or not that would be enough depends on the impedance, Amps and voltage of your load. (not an expert on this)
If you can spare the PCB space and a few extra diodes i would be safe and add one anyway, there's all kinds of horror stores about design faults found after the board hit production.