Try your regex with a tool like http://jsregex.com/ (There is many) or better, a unit test.
For a naive validation:
function validateDate(testdate) {
var date_regex = /^\d{2}\/\d{2}\/\d{4}$/ ;
return date_regex.test(testdate);
}
In your case, to validate (MM/DD/YYYY), with a year between 1900 and 2099, I'll write it like that:
function validateDate(testdate) {
var date_regex = /^(0[1-9]|1[0-2])\/(0[1-9]|1\d|2\d|3[01])\/(19|20)\d{2}$/ ;
return date_regex.test(testdate);
}