I doubt that he would. Illustrious though the
House of Stuart may be by modern standards, its rule was a mere blink of an eye beside that of the ruling dynasties of Gondor. Not only was Gondor much older by Denethor's day than either Scotland or the rest of Britain was under the Stuarts, but also Denethor's own house had ruled Gondor far longer than any dynasty has ruled Britain. To put this into its proper perspective, Gondor was established in SA 3320, and had therefore existed for a little over 3,100 years by the time of the War of the Ring. The stewards ruled it from the death of Earnur in TA 2050 until the accession of Elessar in TA 3019, or a little over 960 years. By contrast the Stuarts ruled Scotland from 1371 a.d. (and England from 1603 a.d.) to 1714 a.d.
Aside from the obvious seniority of his house, Denethor was entirely convinced of the superiority of Gondor over all other societies and cultures. The Stuarts (and, indeed, every other ruling house of Europe since the fall of Rome) would have seemed to him nothing more than barbarian chieftains, aping the true civilisation represented by his own people.
You're right about the two houses having something in common, though: the name 'Stuart' derives from the Old-English word for a steward. According to the source to which I've linked above, the royal house of that name appears to have originated in eleventh-century Brittany as stewards to the counts of Dol. Possibly this is a little historical joke on Tolkien's part.