Update Row Cell values in GridView on DropDownList SelectedIndex Changed in ASP.Net

Here I have created sample that full full your requirement.

HTML

<asp:GridView ID="GridView1" runat="server" AutoGenerateColumns="false" OnRowDataBound="OnRowDataBound">
    <Columns>
        <asp:BoundField HeaderText="Name" DataField="ContactName" />
        <asp:TemplateField HeaderText="Country">
            <ItemTemplate>
                <asp:Label ID="lblCountry" runat="server" Text='<%# Eval("Country") %>' Visible="false" />
                <asp:DropDownList ID="ddlCountries" runat="server" AutoPostBack="true" OnSelectedIndexChanged="SelectedIndexChanged">
                </asp:DropDownList>
            </ItemTemplate>
        </asp:TemplateField>
    </Columns>
</asp:GridView>

C#

protected void Page_Load(object sender, EventArgs e)
{
    if (!IsPostBack)
    {
        GridView1.DataSource = GetData("SELECT TOP 5 ContactName, Country FROM Customers");
        GridView1.DataBind();
    }
}
protected void SelectedIndexChanged(object sender, EventArgs e)
{
    GridViewRow row = (GridViewRow)(sender as DropDownList).NamingContainer;
    string country = ((DropDownList)row.FindControl("ddlCountries")).SelectedItem.Value;
    string query = "SELECT ContactName FROM Customers WHERE Country = '" + country + "'";
    DataSet ds = GetData(query);
    row.Cells[0].Text = ds.Tables[0].Rows[0].ItemArray[0].ToString();
}

private DataSet GetData(string query)
{
    string conString = ConfigurationManager.ConnectionStrings["constr"].ConnectionString;
    SqlCommand cmd = new SqlCommand(query);
    using (SqlConnection con = new SqlConnection(conString))
    {
        using (SqlDataAdapter sda = new SqlDataAdapter())
        {
            cmd.Connection = con;

            sda.SelectCommand = cmd;
            using (DataSet ds = new DataSet())
            {
                sda.Fill(ds);
                return ds;
            }
        }
    }
}

protected void OnRowDataBound(object sender, GridViewRowEventArgs e)
{
    if (e.Row.RowType == DataControlRowType.DataRow)
    {
        //Find the DropDownList in the Row
        DropDownList ddlCountries = (e.Row.FindControl("ddlCountries") as DropDownList);
        ddlCountries.DataSource = GetData("SELECT DISTINCT Country FROM Customers");
        ddlCountries.DataTextField = "Country";
        ddlCountries.DataValueField = "Country";
        ddlCountries.DataBind();

        //Add Default Item in the DropDownList
        ddlCountries.Items.Insert(0, new ListItem("Please select"));

        // Select the Country of Customer in DropDownList
        string country = (e.Row.FindControl("lblCountry") as Label).Text;
        ddlCountries.Items.FindByValue(country).Selected = true;
    }
}

Sceenshot

1) On Page Load

2) After changing country Belgium