I have removed the validation group from the Validator and its working fine.
HTML
Default.aspx
<%@ Register TagPrefix="uc1" TagName="TextBoxUserControl" Src="~/WebUserControl.ascx" %>
<form id="form1" runat="server">
<div>
<asp:ScriptManager runat="server" />
<asp:GridView ID="GridView1" runat="server" AutoGenerateColumns="false">
<Columns>
<asp:TemplateField HeaderText="Value">
<ItemTemplate>
<uc1:TextBoxUserControl runat="server" />
</ItemTemplate>
</asp:TemplateField>
</Columns>
</asp:GridView>
<br />
<uc1:TextBoxUserControl runat="server" />
<br />
<asp:Button Text="Validate" runat="server" />
</div>
</form>
UserControl.ascx
<asp:UpdatePanel ID="UpdatePaneltxt" runat="server" UpdateMode="Conditional">
<ContentTemplate>
<asp:TextBox ID="txtTextBox" runat="server" Width="217px" class="round default-width-input"
AutoPostBack="true"></asp:TextBox>
<asp:RequiredFieldValidator ID="rf" runat="server" ControlToValidate="txtTextBox"
Display="Dynamic" ErrorMessage="* Required" ForeColor="Red"></asp:RequiredFieldValidator>
</ContentTemplate>
</asp:UpdatePanel>
Screenshot
