Needs["Calendar`"];
myDay[x_List] := DateString[x, {"DayName"}] /; DateQ[x]
myDay[{2000, 1, 1}]
->"Saturday"
myDay[{2000, 13, 13}]
->myDay[{2000, 13, 13}]
Of course, you can send a message (or Abort [] or something else) if you want:
Needs["Calendar`"];
Clear@myDay;
myDay[x_] /; If[DateQ[x], True, Message[myDay::nodate, x]; False] :=
DateString[x, {"DayName"}]
myDay::nodate = "The argument `1` is not a valid date.";
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