At its simplest I think you need something like this. You may need to extend the range to include the entire paragraph, check the style name etc.
Dim aTable As Word.Table
Dim rng As Word.Range
For Each aTable In ActiveDocument.Tables
If aTable.Rows(1).Cells(2).Shading.BackgroundPatternColor = wdColorGray15 Then
Set rng = aTable.Range
rng.Move unit:=wdParagraph, Count:=1
aTable.Delete
rng.Delete
Set rng = Nothing
End If
Next aTable