I find "yeux doux" translated in several sources as "goo-goo eyes" or "googly eyes" in English, but none as "doe's eyes".

Popular songs in English refer to "making eyes at" or "having eyes for" as a way of showing love.  ("Mom, he's making eyes at me", "I only have eyes for you.")  It's a bit old-fashioned, however, not in everyday use any more.

One can also make "sheep's eyes" in English to express love.

And a beautiful woman can be "doe-eyed".

However, I don't find any reference in English sources to "making doe's eyes", and I've never heard the expression myself.  Perhaps it is a literal translation of a French expression.

An approximate translation of the sentence might be: "Making eyes at your screen all day will get you nowhere.  If it worked we would know it."  It's a challanging sentence to translate into coherent English.

Walter B.