Is there any difference in meaning between the expressions "to fall on someone" and "to fall to someone"? They seem to have the same meaning, namely, that something is someone's responsibility. There may be a difference in usage, though.
"It falls to the Court to verify whether the Member State concerned has adopted appropriate measures for ensuring the free movement of goods."
Would it be wrong to replace "falls to" with "falls on"? I don't think so. What do you think?
To fall on someone can mean to attack someone (although this is becoming obsolete if it hasn't become so already), and to fall to someone can mean to be conquered by someone.
But I agree with the meaning you ascribed to the phrase in your example. It falls to the court means it is the task of the court. I think it would sound less natural to replace it with it falls on the court.