Disable DataGrid current cell border in FullRow selection mode

You could set the BorderThickness for DataGridCell to 0

<DataGrid ...
          SelectionUnit="FullRow">
    <DataGrid.CellStyle>
        <Style TargetType="DataGridCell">
            <Setter Property="BorderThickness" Value="0"/>
            <!-- Update from comments.
                 Remove the focus indication for the selected cell -->
            <Setter Property="FocusVisualStyle" Value="{x:Null}"/>
        </Style>
    </DataGrid.CellStyle>
    <!-- ... -->
</DataGrid>