AFIK, there is no official resource for finding out the disabled dates in jQuery Datepicker.
However jQuery UI Datepicker is a library upon Javascript, there is always a workaround.  Below is a workaround I have tried.
Step #1:
Below is a way to disable a list of dates in jQuery Datepicker.
var array = ["2014-03-14", "2014-03-18", "2014-03-16", "2014-04-01"]
$('input').datepicker({
    dateFormat: 'yy-mm-dd',
    beforeShowDay: function (date) {
        var string = jQuery.datepicker.formatDate('yy-mm-dd', date);
        return [array.indexOf(string) == -1]
    },
    onSelect: function (date) {
            console.log(findNextDisabledDateWithinMonth(date));
    }
});
Now here my approach to get the next disabled date within the month.
Step #2:
function findNextDisabledDateWithinMonth(date) {
    var currentDate = Number(date.split('-')[2]);
    var month = $('.ui-datepicker-title>.ui-datepicker-month').text(); //Number(date.split('-')[1])
    var year = $('.ui-datepicker-title>.ui-datepicker-year').text();  //Number(date.split('-')[0])
    var nextConsectiveDates = [];
    $.each($('.ui-state-disabled').find('.ui-state-default'), function (i, value) {
        var numericDate = +$(value).text();
        if (currentDate < numericDate) {
            nextConsectiveDates.push(numericDate);
        }
    });
    var nextDisabledDate = nextConsectiveDates[0] + "-" + month + "-" + year;
    return nextDisabledDate;
}
JSFiddle
NOTE: This is work only for the selected date's month
Approach #2
As @ Salman A mentioned in his comment, I feel the best way is to go with it.  Since my approach will is constrained within month.
Here is an elegant approach to solve your problem.
var array = ["2014-05-01", "2014-04-14", "2014-04-18", "2014-04-16"];
// Convert all those string dates into Date array
var arrayAsDateObjects = convertStringToDateObject(array);
$('input').datepicker({
    dateFormat: 'yy-mm-dd',
    beforeShowDay: function (date) {
        var string = jQuery.datepicker.formatDate('yy-mm-dd', date);
        return [array.indexOf(string) == -1]
    },
    onSelect: function (date) {
        alert(findNextDisabledDateWithinMonth(date).toDateString());
    }
});
//To find the next exact disabled date based on the selected date
function findNextDisabledDateWithinMonth(date) {
    var splitDate = date.split("-");
    var selectedDate = new Date(splitDate[0], Number(splitDate[1]) - 1, splitDate[2]);
    var nextDisabledDate = null;
    $.each(arrayAsDateObjects, function (i, ele) {
        if (selectedDate < ele) {
            nextDisabledDate = ele;
            return false;
        } else {
            nextDisabledDate = "No Disabled dates available";
        }
    });
    return nextDisabledDate;
}
//convert all the string dates to Date object
function convertStringToDateObject(array) {
    var ls = [];
    $.each(array, function (i, ele) {
        var splitDate = ele.split("-");
        var date = new Date(splitDate[0], Number(splitDate[1]) - 1, splitDate[2]);
        ls.push(date);
    });
    // Sort the dates in ascending order(https://stackoverflow.com/a/10124053/1671639)
    ls.sort(function (a, b) {
        return a - b;
    });
    return ls;
}
for sorting dates, I took reference from @Phrogz 's answer