I am a chemist, and since this appears to be a serious question (despite its humor), I thought I would offer a comment.
Briefly --- the vomiting center is located in the medulla oblongata, located in the rear part of the brain. This portion of the brain acts on information supplied by the stomach, the intestines, the gag reflex in the throat, the inner ear, and most importantly, the "chemoreceptor trigger zone" (CTZ), which is located on the floor of the fourth ventricle in the brain (in case anyone is interested). The CTZ takes in data about the possible presence of poisons or toxins in the blood and alerts the vomiting center to do its thing when it believes the body has been poisoned.
Stomach acid (or hydrochloric acid), in undiluted form, has a pH of roughly 1.0, and would indeed destroy one's rope if it were to make contact with it in this pure and undiluted form. However, in an important first step as the vomiting process is initiated, the vomiting center triggers the small intestine to send a fair portion of its contents back into the stomach. This starts approx. one minute before vomiting and lasts about 45 seconds. This has the effect of neutralizing the vomit*, making it less damaging to the esophagus, throat, mouth, and teeth. Some researchers contend that expulsion of the intestinal content is a necessary part of the vomiting act in itself (since the intestine is where the toxin was detected by the CTZ), and that the movement into the stomach puts it in place for being ejected. The transfer of intestinal matter also increases the volume in the stomach, which makes vomiting easier.
So, logic would dictate that if the actual pH of vomit (as opposed to the pH of stomach contents prior to combining with material from the small intestine) were the same as that of the hydrocholoric acid used in digestion, then the esophogus would be severely damaged as a result of vomiting (the esophogus has no protection against hydrochoric acid as does the stomach). Obviously, people do not routinely suffer damage to the esophogus as a result of normal vomiting of digested food. So, although the pH of vomit is probably slightly acidic, I have my doubts that it is sufficiently acidic to damage your rope (especially if you cleaned or washed it in a reasonable amount of time following contact).
More importantly, there are two factors that one should be concerned about that dictate whether or not a rope should used or retired (with vomit or otherwise). The first is its actual physical condition, and the second is your confidence in its ability to hold a fall. Certainly, if the rope does not pass the first criteria, then the second criteria is mute. However, if the rope appears to be good, yet you do not have absolute confidence in it (for whatever reason), then it should be retired. Climbing when you have anxiety about your gear can be as dangerous as climbing with a less than adequate rope.
So, the upshot is: I doubt your rope is damaged. BUT, if YOU think it is suspect, then retire it.
Sorry for the long post.
regards,
Brad Seabourn, Ph.D.
Manhattan, KS
=========
*Ivan M. Lang & S.K. Sarna, "Motor and Myolectric Activity Associated with Vomiting, Regurgitation, and Nausea," in Wood, J.D., ed., Handbook of Physiology: The Gastrointestinal System I: Motility and Circulation, Bethesda, Md.: Bethesda, Md., Physiological Society, 1989: 1179-1198.