Are flyback diodes needed for this op-amp voltage controlled inductor?

I would still add diodes between the output of the opamp and both supply rails.

Such diodes are already present anyway inside the opamp (for ESD protection) but you do not want to damage these diodes as then you'd have to replace the opamp.

You're right that under normal working conditions the current through the coil is not interrupted (which is the cause for the high damaging voltage) but what about when you switch the circuit on, off or a glitch occurs on the supply ?

So I would not take the risk and just add reverse-biased diodes to protect the opamp's output. I'd use fast Schottky diodes which can handle 1 A forward current (this is just my guess).

When you try to instantaneously change the current from 6 amps to 3 amps the left side of the inductor will react in a way to keep the current flowing at 6 amps (just for a short time until the stored energy is lost somewhere). So, its only option is to generate a voltage that is high enough to force current through the upper transistor in your push-pull stage inside the op-amp (or generate a spark).

Given that this top transistor is never actually intentionally turned on in either scenario, the inductor will pretty instantaneously generate a bigger and bigger voltage until that transistor breaks down or you get a spark somewhere.

It seems to me that a protection diode is needed.

Often when you have a discrete push-pull circuit you don't need flyback diodes, but they will do no harm to put in, and if you can't analyze the (perhaps integrated circuit) amplifier output section of the amplifier to determine they are not necessary this is a good (and inexpensive) practice. In this case you can only sink current, so a diode to the +10 supply would be called for. A Schottky diode is much safer than a regular diode since little current will flow through internal junctions when the Schottky conducts in parallel. Current flowing through internal junctions could cause issues that are difficult to predict.

Note that there is still something to worry about- if there is such a diode (or the equivalent) and if the 10V supply is suddenly disconnected, the collapsing field may cause the supply voltage to the chip to rise to a high enough level to damage the chip (depending on bus capacitance etc.). So a Zener or a TVS capable of absorbing the energy in the coil may be a good idea- put it from +10 to ground, or from the amplifier output to ground.