Fat in Milk Increases Stomach Acid
Interestingly, drinking milk may temporarily buffer stomach acids to make you feel more comfortable, but the fats in milk will actually stimulate the stomach to produce more acid.  Drinking an eight-ounce glass of whole milk containing eight grams of fat is nearly equivalent to eating a single serving of Lay’s potato chips at ten grams per serving. 
Foods high in fat should be avoided by anyone that suffers from acid reflux. Fatty foods take longer to digest, spending more time in the stomach, which requires the production of more stomach acids to move things along. 
Foods high in fat, such as full fat milk, may also relax the lower esophageal sphincter (LES), the muscle between the stomach and the esophagus that keeps stomach acids contained, causing it to open and allow foods and gastric acids to reflux into the esophagus. 
Too Much Milk Can Cause Acid Reflux
Eating too much is actually the leading cause of acid reflux episodes. When the stomach is too full, it distends which puts pressure on the LES, which can cause it to open. 
Alternatives to Whole Milk for Acid Reflux
The prescription for managing acid reflux without giving up milk is to find a less fatty alternative like skim milk. There are several other lower fat milk alternatives such as soy (3.5g), almond (2.5 g), flax (2.5g), and rice milk (2.5 g). 
Study Reveals Infant GERD May be Cow’s Milk Allergy
A 2001 study revealed that cow’s milk is also bad for acid reflux experienced in infants.
GERD is a common condition in infancy and a study published in the journal Gut and Liver concluded that cow’s milk may be to blame in some cases. The researchers studied eighty-one infants with symptoms of GERD a proton-pump inhibitor commonly used in the treatment of GERD. The researchers found that though two-thirds responded positively to the treatment, the remainder did not. For these patients, cow’s milk was removed from either the diets of the infants or the diets of the breast-feeding mother. Four weeks following the removal of cow’s milk from the group’s diet, all signs and symptoms of GERD were resolved. The findings reveal that infant GERD may actually be cow’s milk allergy (CMA). 
The American Academy of Pediatrics warns against giving infants one-year-old and under milk for a variety of health reasons. First, the proteins and fats present in cow’s milk are difficult for an infant’s undeveloped system to handle. Also, cow’s milk does not provide enough iron, vitamin E or essential fatty acids, while containing too much protein, potassium, and sodium.
The AAP recommends breast milk or iron-fortified formula for the first twelve months of life.