According to this article from The Atlantic,, yes, he could claim that he declared them declassified and no paperwork would have been necessary. However, LegalEagle, in this video at 12:12, says that documents must be treated according to their classification markings (regardless of the actual classification), in which case there would be "paperwork" (albeit just that he would have to change or have changed the classification markings). But it's not clear that this actually must apply to the president who (with exceptions below) has ultimate authority over classification.
Either way, not all documents can be declassified by the president: certain types of documents, such as those giving the identities of spies or containing nuclear secrets, have their classification and declassification controlled by statute, not by the president. Further, the president's ability to declassify documents ends when he leaves office, and any documents he declassified documents may be reclassified again by those in the government authorised to do so.
The description from The Atlantic:
The 1988 Supreme Court case Navy v. Egan confirmed that classification authority flows from the president except in specific instances separated from his powers by law. And here is where things get theological: A president can make most documents classified or declassified simply by willing them so. This peculiar power is so great that the government has an office that exists solely to manage it: the Information Security Oversight Office...
...J. William Leonard, led the office under George W. Bush, and he confirmed the lack of general limitation of his boss’s power. While a president is president, Leonard told me, “the rules and procedures governing the classification and declassification of information apply to everyone else.” And that means Trump could have declassified whatever he wished (again, with specific limitations soon to be discussed) before carting it off to Mar-a-Lago. He would not have had to file paperwork—just “utter the magic words,” Leonard told me. He could have waved his hand over the U-Haul trailer as it headed out the White House driveway and down I-95 toward Florida, and there would have been no classified material in there to mishandle.
Leonard noted important caveats, however. First, Trump’s power to declassify ended with his presidency. Second, that U-Haul could be reclassified by someone else. (Depending on traffic and the sharpness of the Biden administration, I would imagine it could have been reclassified somewhere around Fredericksburg, Virginia.) And third, there are certain materials that presidents cannot classify and declassify at will. One such category of material is the identity of spies.
Another is nuclear secrets. The Atomic Energy Acts of 1946 and 1954 produced an even stranger category of classified knowledge. Anything related to the production or use of nuclear weapons and nuclear power is inherently classified, and Trump could utter whatever words he pleased yet still be in possession of classified material. Where are our nuclear warheads? What tricks have we developed to make sure they work? This information is “born secret” no matter who produces it. The restrictions on documents of this type are incredibly tight. In the unlikely event that Trump came up with a new way to enrich uranium, and scribbled it on a cocktail napkin poolside at Mar-a-Lago early this year, that napkin would instantly have become a classified document subject to various controls and procedures, and possibly illegal for the former president to possess. Of course if he did so, no prosecutor would pursue him. A certain amount of leeway is crucial to the system.
The description from the LegalEagle video:
We also don't know if trump followed the procedure for declassifying the
documents. He can't just think declassification in his head while president
and the documents are declassified. The lawyers at Just Security summarize
the process for declassification like this: classified documents have
classification markings in the header and footer of each page indicating
the level classification for the document as a whole. Furthermore
classified records have cover sheets that specifically indicate when the
record was classified by whom and under what authority as well as when the
classification expires. If Trump did in fact order the declassification, he
still needed to make sure his staff took the necessary next steps to modify
the classification markings on the documents before he could actually
handle and store the records as a private citizen [emphasis mine --cjs]
as if they were unclassified. Under security classification rules a
classification marking on a document has to be treated as valid and binding
unless and until a subsequent marking replaces it. Appropriate government
staffers would have needed to cross out the classification markings in the
headers and footers and stamped "declassified" on the record noting when it
was declassified by whom and under what authority.
Since that does not appear to have been done with the classified documents
reportedly identified to date, the documents remain classified and had to
be treated as classified for handling and storage purposes.
The New York Times article "Presidential Power to Declassify Information, Explained" directly asks whether the president must follow the procedures:
Do presidents have to obey the usual procedures?
There is no Supreme Court precedent definitively answering that question.
Proponents of a strong view of presidential power have argued in other contexts that presidents are not personally bound by the rules and procedures that regulate the conduct of their subordinates in the executive branch — and that presidents can even disregard executive orders without first rescinding them. Others disagree with that vision of executive power.
There have been several comments (some with many upvotes) essentially saying that it doesn't make sense or is a bad thing that we have nothing (neither statute law nor a court ruling about executive orders or executive power) clearly stopping a president from declassifying documents without actually having at least one copy re-marked as unclassified. (See, e.g., Trilarion, Lag and Joe W, PoloHoleSet, computercarguy, and Joe W again.) This is certainly not an unreasonable opinion, and "secret" declassifications can clearly cause problems, but that does not change the current state of the law and regulation. Absent a court judgement about a president declassifying documents in this way, it is not possible to say that it can't be done, no matter how terrible an idea you think it is.
Also note that "bad ideas" like this are not uncommon in law. For example, members of the Trump administration appear to have violated various laws that only Trump can enforce, and he chose not to enforce them. Those laws were obviously not well drafted to handle situations like this, but they are what they are.