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I'm trying to learn how to select the correct output transformers to match the load impedance of my 23 ohm headphones.

So the output transformers are matching "line matchers".
The turns ratio is 5.1 with 15K ohms/600 ohms.
I'll use a matched pair (left/right) in my tube amp build.

Please correct me but my understanding is that
5.1 squared = 26.01
so then i multiply
26.01 x 23 ohms = 598.23 ohms
so then 598.23 is close enough to 600 ohms which means the above output transformers are perfectly matched to the 23 ohm load of my headphones.
Right?

It would be more correct to divide 600 ohms by 26.1 getting 22.99 ohms. However if you connect the secondary of the line matching transformer to your headphones, the impedance on the primary will be affected by the same amount. 15k divided by 26.1 gives you 574.7 ohms. This is probably not going to be a suitable load impedance for the tube circuit.

What tubes and output type is your amp? Normally I would start with the ideal primary impedance for the tubes and then search for a transformer that would give me close to 23 ohms on the secondary.