Does Coke cure acid reflux?

Not for most people, no.

GERD (gastro esophageal reflux disease), stomach contents passing into the esophagus, is usually not cured with medications. The factors that contribute to GERD are the lower esophageal sphincter tone, hiatal hernias, esophageal contractions, and emptying of the stomach. To cure it you'd have to have surgery to reinforce the lower esophageal shpincter.

Most people find that the symptoms of chest pain, heartburn, throat discomfort/ hoarseness and swallowing problems can be treated by decreasing the acidityof the stomach contents with medications such as Zantac (ranitidine) and Prilosec (omeprazole). Usually after a short course the symptoms resolve and the medication can be stopped. Most people find that acidic foods such as tomato sauce, citrus and carbonated beverages make it worse. In additions large meals, fatty meals and lying down with food in the stomach increase the reflux and so make it worse. Acidic foods, alcohol and NSAIDs (Aspirin, ibuprofen, Aleve, naproxen) injure the stomach lining and so make symptoms worse. Caffiene (as is found in Coke) relaxes the lower esophgeal sphincter and so increases the amount refluxed.

There is a theory, held by a small number of people and virtually no well respected experts, that the symptoms are from insufficient stomach acid and so taking acids in the form of apple cider vinegar (and I guess Coke according to the questioner) will help. If this was the case then one would expect the medications like Tums and Zantac to make thing sworse, not better.

While it is possible that there is a small group of people who have the opposite from the more common problem, for the vast majority of peope with stomach problems vinegar and soda should be avoided.

Soda is very acidic and would probably be a poor choice for someone with acid reflux. You'd be much better off with water so you don't get all of that acid, sugar, and empty calories anyway.