He should either call the troops out or shut up.

A illegal inquiry would be a coup.

And therefore, giving the President the benefit of the doubt and taking him at his word, we have a constitutional crisis and, I reiterate, he should declare martial law.

He would need a strong justification for that, that doesn't now exist. Nor would it actually do anything to solve sthe problem at hand. Martial law is more like what Putin would do to disappear his critics.

You just said it yourself: a "coup." Whether he actually believes that or not is irrelevant to my point: I'm calling his bluff. If he claims the impeachment inquiry is an attempt to unlawfully depose him, then he should put his money where his mouth is and take action to counter it. So if martial law isn't an appropriate response to an (alleged) attempt to illegally remove the President from office, then what is?

Well as this coup is not one that uses force, any use of force to counter it will be used as justification to remove Trump for cause. In this manner it could even be considered a trap in that Trump fighting against it even in the legal system being used to justify it.

This is the type of thing that requires the checks and balances within the legal system to be used. The basic information of what is and has happened needs to be disseminated widely and any actual wrongdoing by the parties against Trump are them to be meet with the full force of the law included voting them out of office or with illegality throwing them in jail.

This of course is one of the reasons that an honest media is a good thing rather than the smear merchants and fake news mongers that are wide spread now and an actual set of rules to be followed in the House of Representatives.

Point of clarification: I'm not agreeing with him, merely playing devil's advocate.
Yes, checks and balances, which, by Trump's logic, are apparātūs of the so-called coupmakers.
I mean, if martial law worked for the likes of Erdoğan, maybe by some insane miracle it could work for Trump.