Validate HTML TextArea using RequiredFieldValdiator in ASP.Net

You cannot. You will need to use CustomValidator.

HTML

Message:  
<textarea id="textarea11"></textarea>
<br />
<asp:CustomValidator ID="CustomValidator1" ErrorMessage="Please enter some message"
    runat="server" ClientValidationFunction="MessageValidation" />
<br />
<asp:Button ID="Button1" Text="Save" runat="server" />
<script type="text/javascript">
    function MessageValidation(sender, args) {
        var message = document.getElementById("textarea11").value;
        if (message == "") {
            args.IsValid = false;
        }
        else {
            args.IsValid = true;
        }
    }
</script>

Screenshot