I have to disagree with Rabbi Singer on this issue.
Responsa of the Rambam (Maimonides) 448
(To R' Ovadia the Convert) A question regarding these Yishmaelites (Muslims) that you said, they are not worshiping foreign worship (Avoda Zara) and your Rabbi said to you, that they are worshiping foreign worship and the stones they toss at their false holy place (called by an embarrassing nickname) are to the Markolis idol. And he responded to you improperly until you were saddened and ashamed and he said about you the verse, "answer the fool according to his foolishness".
Rambam admits the Muslims do Peor, Kemosh, and Markolis rituals at their fake holy place but offers excuses why not to hold the Muslims guilty of idolatry. In the same answer, Rambam also states straight out that is too dangerous for him to put in print some of the stupid beliefs of the Muslims, but calls them monotheists.
Rabbi Singer seems to be siding with the Rambam, disregarding Rambam's overt statement, he was not free to write about the faults of the Muslims.
Other big Rabbis who were under less pressure than the Rambam held that the Muslims were guilty of idolatry by merely doing the Mecca rituals, regardless of their Monotheistic beliefs.
Sefer HaEshkol brings the opinion of one of the Gaonim, that at this time (he lived at the time of the Muslims) the Yishmaelites are guilty of idolatry at their fake holy places, even though they do not recognize they are doing idolatry.
Meiri also brings the sages of Spain that hold the Muslims are guilty of idolatry due to their rituals at their fake holy places.
In his commentary to tractate Avoda Zara 64b Meiri leans towards that view point.
Rashbatz brings an uncensored version of the Kuzari that accuses the Muslims of idolatry because of their stone throwing ritual at their fake holy place.
Regardless of this question, for other reasons the Muslims are guilty of Avoda Zara, but I am not planning to write a major essay on this here. Instead I urge you to research the subject.