No.
For one thing, there is a sphincter at the junction between the esophagus and stomach that’s normally tightly constricted to keep stomach contents from re-entering the esophagus (except in cases like GERD). Here’s what that sphincter looks like by endoscopy.
For another, the esophagus doesn’t enter the stomach at the very top. The upper dome of the stomach, called the fundus, encloses a fairly substantial space higher up than the esophagus in most people. If you took a drink or ate a bit of food and then stood on your head, and if that stomach content then fell by the force of gravity, it would more likely fall into this space than up the esophagus.
Here’s an upside-down stomach. You can see where the food or drink would fall in a person standing on their head, especially with that sphincter blocking the way into the esophagus.