You can wrap ScrollViewer into Grid, and bind scrollviewer's Width and Height properties to grid's ActualWidth and ActualHeight. So the scrollviewer will have fixed size equal to the size of the grid which will change when window resize.
Example:
<Grid x:Name="LayoutRoot" Background="White">
<Grid Background="#FFF1F1F1" Height="49" VerticalAlignment="Top">
<Button Content="Обзор" Margin="0,13,175.25,0" VerticalAlignment="Top" FontSize="14.667" HorizontalAlignment="Right" Width="95.147">
</Button>
<Label Content="{Binding DocPath, Converter={StaticResource FileNameConverter}, FallbackValue=Выберите файл, TargetNullValue=Выберите файл}" HorizontalAlignment="Left" VerticalAlignment="Top" Width="342.603" Margin="10,10,0,0" Height="33"/>
<Button Content="Загрузить данные" HorizontalAlignment="Right" Margin="0,13,10,0" VerticalAlignment="Top" Width="151.147" FontSize="14.667">
</Button>
</Grid>
<Grid x:Name="scrollBorder" Margin="10,54,10,10">
<ScrollViewer x:Name="LogScroller" VerticalScrollBarVisibility="Auto" HorizontalScrollBarVisibility="Auto"
HorizontalAlignment="Left" VerticalAlignment="Top"
Height="{Binding ActualHeight, ElementName=scrollBorder}" Width="{Binding ActualWidth, ElementName=scrollBorder}" >
<ItemsControl ItemsSource="{Binding Log}" />
</ScrollViewer>
</Grid>
</Grid>