...the surgeon that was ordered to dress the captain was a tall, thin man, called Weeks, of Warwick; very soon after we got on board the schooner, the men's hands belonging to the schooner was tied behind their backs, and put in boats and put on shore.  ...

A list of five men's names, that was concerned in destroying his Majesty's schooner Gaspee:
John Brown and Joseph Brown, principal men of the town of Providence; Simeon Potter of Bristol; Doctor Weeks, of Warwick; Richmond, of Providence.

And thus, My Lord, this forced confession of the negro Aaron has been held up by the marine, as a hopeful and sure clue to unravel this mystery of iniquity. The fellow might probably have heard the names of the most noted and principal traders at Providence, and other trading towns in that neighborhood, though he might not know their persons; and one Dr. Weeks, inserted in his list, he might know personally, as an attendant upon his master's family.

...that before they went ashore, a doctor, whom they called Weeks, from one of the boats, dressed the Captain's wounds; that when they had landed the people, they untied their hands and let them go, and the captain of the schooner they carried up to a house; ...ne

In reviewing Montagu's list of possible suspects, he named a "....Doctor Weeks of Warwick, and a Richmond of Providence." This could be interpreted that he meant both Weeks and Richmond were doctors.  In researching Richmond, we can find no good evidence that Dr. Benjamin Richmond of Little Compton would be involved in the raid on the Gaspee, but it remains a remote possibility. As a wild postulation, it could be that Dr. Benjamin Richmond from Little Compton (or an unknown Dr. Weeks from Warwick) took on medical student John Mawney for some clinical preceptorship, and that both gentlemen attended to Aaron Briggs masters, Captain Samuel Tompkins or his elderly father-in-law Samuel Thurston, on the not-too-distant Prudence Island.  Here, Aaron Briggs would have seen the doctors and somehow got the idea that one of them was a Dr. Weeks. But again, this is wildly speculative, and is not borne out by any facts.  Another physician involved was Dr. Henry Sterling of Providence, who tended to Dudingston's wounds while he recuperated in the Pawtuxet home of Joseph Rhodes. But it is doubted he was along for the actual attack; elsewise, he would have been the one called on to immediately treat the Lieutenant's wounds aboard the Gaspee, and not John Mawney.

In 2017 we discovered the following discussion while researching Pawtuxet born Revolutionary surgeon- physician Dr Zuriel Waterman. From: Manfred J. Waserman "Dr. Zuriel Waterman: An Itinerant Surgeon in the Revolutionary Era". Proceedings of the American Philosophical Society, Vol. 117, No. 5 (Oct. 25, 1973), pp. 388-403

 The earliest record of Zuriel revealed in his papers is a lone entry in his ledger for the year 1775, indicating that he received four weeks of schooling in navigation. His only entry for 1776 concerns the purchase of a horse for the price of ten pounds and ten shillings, and the next ledger entry, dated August 8, 1777, shows that he paid Dr. Robert Wickes thirty-seven pounds and ten shillings for "my instruction in the medical art." [4]

4. It is possible that only twenty-seven pounds more than the price of a horse gave Zuriel the knowledge and privilege to practice medicine. Wickes does not appear in any of the standard medical biographical works. Usher Parsons, "History of the Medical Profession in Rhode Island," Transactions of the Rhode Island Historical Society 1 (Providence, 1859) mentioned a "Robert Weeks" in Cranston (p. 40). This probably is Wickes. Notices by one Robert Wickes appeared in the Providence Gazette for 1786.

Depending on how one pronounces "Dr Weeks", this is likely our man.  He was practicing in the greater Providence area, but we do not know if in 1772. 

From: Cranston: A Historical Sketch by J. Earl Clauson, (1904), p. 23

During the early years of the Revolutionary War there was considerable suffering from smallpox, which at times spread with such rapidity as to engender a suspicion that physicians disseminated the disease to further their own selfish ends. Captain Edward Knight was given permission in January, 1778, to open his house an as inoculating hospital, he having agreed to receive patients and provide firewood at an expense of twelve shillings for each one, and Dr. Robert Wickes was engaged to attend these patients for a compensation of eight dollars each.

We have that Dr. Robert Wickes was born of Robert Wickes and Margaret Barton, married a Mary Harris (b.1750).  and died in Cranston at age 40. Dr Wickes had a younger sister Sarah Wickes (1742-1820) who married Thomas Greene of the prominent Greene family of Kent County.

The only interesting reference we have found regarding Dr. Weeks is a scrap of paper sold on eBay in May 2005, to wit:

What we might assume this to be is a loan note written by a Dr. Weeks of Cranston who was loaning seven schillings to a relative, also a Dr. Weeks. 1766 was six years before the burning of the Gaspee.

A search of Gendex.com, Whipple.org, LDS , Ancestry.com,and Google.com all are negative for a "Doctor Weeks" or a "Dr. Weeks" or a "Weeks, MD" that would be present in Rhode Island during 1772.

Per the RI Historical Cemeteries Database, there are no possible matches with a Weeks present during 1772.

On the other hand, the USGenWeb listings of the 1790 Federal Census for Warwick, RI gives us five choices:

Weeks, Barney                              *    *    *      6       *
Weeks, Harry                                 *    *    *      3       *
Weeks, John                                  2    1    1       *       *
Weeks, Stently                              1    2    9       1       *
Weeks, Thomas                            2    1    1       *       8

Weeks, Abel                                  1    3    3       *       *
Weeks, Benedict                           2    2    3       *       *
Weeks, Ichabod                            1    2    1       *       *
Weeks, Joseph                              2    3    3       *       *