Is a flyback diode necessary with a TC1412?
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 


"as an H bridge" or 
"to drive an H bridge". 


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:


(1) Yes. Some form of protection is necessary in the general case if using as a motor driver directly. For motor currents under 500 mA "you will get away with it" but would be advised to provide formal "snubbing" in some manner. For motor currents over 500 mA the datasheet suggests "there will be problems". 
(2) No, protection is probably not strictly necessary if you are using the IC to drive MOSFETS which in turn are being used in an H bridge - but it is still "a good idea"[tm] - see below.


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-


A Reverse polarity zener at the MOSFET gate to dissipate Miller capacitance energy coupled from the load (and anything else over spec that may turn up at the gate in real life). If you are driving the MOSFET without drive resistors from the TC1412 it would in most cases handle this sort of transient OK, but ... .
A reverse biased small Schottky diode connected G-S and mounted right at the MOSFET. Stops oscillations when things get out of hand. Again, the IC will notionally handle this and in most cases it's thus not needed.   




Please add a data sheet reference when asking questions related to a specific IC - I've added one to your question.