Handling pushback from direct about compensation review

I've recently been dealing with pushback from a direct who felt that his pay raise was insufficient. (We work in a geographically distributed environment where he works day to day on the client site, so much of this was being handled through email.)

I explained some of the constraints that the company is under when determining what level of a pay adjustment we can give for an employee.

He pushed back again with arguments for why he should get more, and claimed that my line of reasoning was ill-founded and unfair.

I repeated the rationale for the company's decision and said that the change is what it is.

He pushed back one more time, comparing his pay against others and previous roles he has been in.

I told him that there was no negotiation and that the constraints we face had not changed since the start of the dialogue, nor would they change after this response to him.

He has not replied since.

Sigh. This has been handled for now, but it was frustrating and I cointuously questioned myself as to whether I was handling it in the best way possible. Ideally I would have nipped his protests in the bud. Is there guidance in the podcasts that address these types of situatiaons? I haven't found it on my own.

I'd be interested in anyone's hard-won wisdom in this area.

Thanks,
Doug