Yes. However, he would really have no reason to do so, since any information he needs is provided to him via the Secretary of Defense and National Security Council.
It’s an interesting question. On one hand, nobody is going to stop him. On the other hand, mere mortals wanting access to that information, besides having a documented “need to know”, are subjected to a background investigation of essentially unlimited scope.
I wonder at what point in the hierarchy they say, “Sorry, it doesn’t matter whether you are the Under-Under Secretary of Defense for Mess Hall Services (or whatever), you still need a clearance”?
It’s doubtful that anyone could achieve any significant position of authority in DoD—even the UUSD (MHS)—without a clearance.
For that matter, the vast majority of military personnel and DoD civilian employees have to get a security clearance. There are certainly organizations and job descriptions within DoD that are exceptions to this…but the above is true in most cases.
So there’s no chance of someone in the opper levels of management in the Pentagon not having a Top Secret clearance as a necessary requirement of their job.
Some people in the Government, like members of Congress, and the President, are given automatic clearances by virtue of their position.
Clearances are nice, but the “real” secrets are generally in the SCI (Sensitive Compartmentalized Information) system, which is basically a ‘need to know’ pigeonholing system. There are even categories of information that are traditionally explicitly NOT given to the President - for example, the identity of highly placed or sensitive foreign sources or agents. I’m not saying that a President couldn’t insist on seeing this information, but s/he would be definitely and explicitly dissuaded. There are many publicly recorded instances of a President being told “you don’t really need to know that, sir.” Generally (at least in the publicly known cases), the President concedes the point.
I wouldn’t expect any major battle between the President and DCI (or more likely, between the politically-appointed DCI and the suspicious ‘career’ staff) to be too well documented publicly, but stories abound. IIRC, the bestselling “Veil: The Secret Wars of the CIA” (1987) by the Pulitzer Award winning Bob Woodward alluded to a few such incidents. It certainly highlighted the fact that the career CIA staff was often not just suspicious of, but downright obstructive toward the President and their own Director (both of whom were often viewed as temporary political office-holders, at best tangential to ‘the real job at hand’)
The intelligence-gathering and operations side of any intelligence agency are, almost by definition, “breaking the rules” - and not just the laws of the places they operate (even when we have similar laws at home). We’re all aware of instances where they have violated laws and morality by essentially any standard outside their own private opinions. That’s probably inescapable, given the nature of their business, but it creates a certain culture and sense of prerogative that doesn’t always make them eager to open up the books to anyone.
I can assure you that there was little reverence in the intelligence community for either the Office of the President or its holder during, say, the Carter Administration. Presidential photos (issued to every Federal office) were sometimes used as dartboards.