It's no secret that anti-catholic sentiment was very strong in America for most of its early history and well into the 20th century. The first Catholic Presidential nominee did not appear until 1928 in the form of Al Smith, and it was not until JFK that a catholic won the office, with both of them seeing very real opposition from voters based solely on their faith.

However, that's going off on a tangent that as much as I'd like to delve into, is not what you asked about.

As far as I am aware, there is no recorded instance of Washington expressing any firmly anti-catholic views either during the Revolutionary War or during his time in office. However, that is not to say he did not have some views of "Papists", typical of early protestant Americans, that he kept private.

As I said, I could not find any quote from Washington pertaining to the subject, however, in his correspondence with his friend, Irish politician Edward Newenham during his time in office, Newenham does not seem to feel the need to hide his disdain for Irish Papists. In a letter to Washington in 1792, he writes that "the Irish Papists are the most bigotted & adverse People, in the world," and that "they were our Enemies in the Emancipation of America—not a Single address or Name came from them on that Cause of real Liberty—they Stiled you Puritans, by way of Derision."

Of course we should not judge someone simply on the company they keep, but this is the best I could do in regards to finding evidence of Washington's personal opinion on the matter.

As to why Washington did not make many (if any) public comments on Catholicism, you must remember that in its early days, the United States was still quite weak and in its war for independence, had found its greatest allies in the Catholic powers of France and Spain, with catholics such as Lafayette having worked closely with Washington during the war. With the threat of Britain still looming after the war, Washington was in no place to criticize the young nation's most potent allies.

Furthermore, by the late eighteenth century, Catholics made up a close to a meager 1% of the total population, with the bulk of them being concentrated in Maryland (which had been a safe haven for English Catholics who faced persecution in other colonies), so the Catholic population, although viewed with disdain by many, was not seen as a threat. It would not be until later waves of immigration in the nineteenth century (along with incorporation of Catholic populations in formerly Spanish territory) that Catholics were seen as a threat to the US politically (Catholics were suspected of owing allegience to the Pope over the US government) and morally (mixed with a healthy dose of racism, catholic immigrants were often viewed as drunks, in addition to existing conflicts between Catholic and Protestant views). This is where you see organizations such as the Know Nothing party and the KKK start to actively oppose Catholicism in the US.

Sources:

Theodore Maynard, The Story of American Catholicism (1960).

“To George Washington from Edward Newenham, 9 January 1792,” Founders Online, National Archives(http://founders.archives.gov/documents/Washington/05-09-02-0247