He saw the Pope “one glistening January day” in 1870. So that would have been Pius IX.
"Gorgeously robed sampetrini approached, carrying an august Personage in a sedan-chair. Teedie, conscious of his Dutch Reformed heritage, hissed frantically that “he didn’t believe in popes—that no real American would.” As [his sister] Corinne later recalled:
The Pope … his benign face framed in white hair and the close cap which he wore, caught sight of the group of eager little children craning their necks to see him pass; and he smiled and put out one fragile, delicate hand toward us, and, lo! the late scoffer who, in spite of the ardent Americanism that burned in his eleven-year-old soul, had as much reverence as militant patriotism in his nature, fell upon his knees and kissed the delicate hand, which for a brief moment was laid upon his fair curly hair.
Source: The Rise of Theodore Roosevelt, by Edmund Morris