Just to confirm... so you don't need to say 'full of milk'? And what if you wanted to say 'the glass is (whole) milk'? I know it's a bit clumsy, but you might say something similar in English. Would that be 'het glas is volle melk'?
That would be: "het glas is vol met melk" "Volle melk" means something different: how creamy it is (don't know how english people call that. Percentage of fat) You can buy: volle melk, halvolle melk, magere melk The french say "lait demí-ecreme" for "halfvol"
To answer the first part of your question: no, we don't need to say 'full of milk'. We can leave the word 'of'
Another clumsy English sentence just so I would fully understand this: Het glas is vol (met) volle melk. would then be The glass is full of whole milk.?