Simple Answer: Nothing.
Background: First of all, I must tell that your statement So why doesn't pepsin acts on the milk proteins? is based on wrong assumption that pepsin does not act on milk proteins.
Milk contains a protein called casein, which has many different types (usually, we talk about $\kappa$-casein). Infants, which are dependent solely on milk for their survival, have a special protein for digestion of casein, known as rennin (aka chymosin). What chymosin does is breaking casein into para-casein and a glycomacropeptide. The reaction is like this:
In general terms, it is said to be a coagulation of milk. It is important to coagulate milk so that milk doesn't pass on to intestine just like water.
Back to the reaction, converting casein to para-casein is all chymosin does. The rest of the digestion of para-casein is carried out by pepsin only, meaning that there is no requirement to prevent pepsin from acting on casein. Also, the quantity of chymosin produced has nothing to do with it. I hope it gets clear to you now :)
PS: the glycomacropeptide is a biologically active protein and has been shown to stimulate the release of cholecystokinin in the body2.
EDIT: Adults too can digest casein with the sole help of pepsin. But pepsin requires lower pH for digesting casein, as low as pH 2.03. At pH higher than this (about 4.0 to 7.0), pepsin gives unusual peptides on digestion4. Infants have a higher pH than this, which makes pepsin incapable of digesting casein in their body5. This also, partly, explains why infants need chymosin for this purpose.
1. Harper’s Review of Physiological Chemistry, 5th edition (p. 177, 1955)
2. Keogh JB, Woonton BW, Taylor CM, Janakievski F, Desilva K, Clifton PM. Effect of glycomacropeptide fractions on cholecystokinin and food intake. Br J Nutr. 2010;104:286–90
3. Qi W, Su R, He Z, Zhang Y, Jin F. 2007. Pepsin-induced changes in the size and molecular weight distribution of bovine casein during enzymatic hydrolysis. J Dairy Sci 90:5004–5011. doi:.10.3168/jds.2007-0230
4. Ono T, Takagi Y, Kunishi I. Casein phosphopeptides release from casein micelles by successive digestion with pepsin and trypsin. Biosci Biotechnol Biochem. 1998;62:16–21. doi: 10.1271/bbb.62.16.
5. Dallas D. C.; Underwood M. A.; Zivkovic A. M.; German J. B. Digestion of protein in premature and term infants. J. Nutr. Disord. Ther. 2012, 21121–9.