Three is no in-built function to allow you to modify the visibility of the calendar icon but you can hack away at the elements using jQuery to achieve the desired result.
Hopefully the below will give you an idea of where to start with this.
$(document).ready(function() {
// Find and remove k-picker-wrap class from Kendo DatePicker 'datepicker'
$("#datepicker_wrapper").removeClass("k-picker-wrap");
// Hide the icon
$(".k-state-default > .k-select").css("visibility","hidden");
// Add a click handler to the DatePicker to replace the functionality of the calendar button
var datepicker = $("#datepicker").data("kendoDatePicker");
$(".k-input").click(function() {
datepicker.open();
});
});
You may have to modify this slightly to meet your needs.