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
