I'm still looking for a way to do this with one Grid, but a partial solution is to use ItemSize to avoid manually sizing Spacings in outer Grid.

(* Eight data triplets  *)
data1 := {{0, -2, 14632971213167*^-14}, {0, -1, -84548187169114*^-14}, 
          {0, 0, -37563603672040*^-13}, {0, 1, 33855169168385*^-13}, 
          {0, 2, -95791963387872*^-14}, {0, 3, 15772038513228*^-14}, 
          {0, 4, -16616417199501*^-15}, {0, 5, 81214629983568*^-17}}

(* Create a table of the data w/ header row *)
tbl01 := Prepend[
  Table[{i, data1[[i]][[1]], data1[[i]][[2]], 
    NumberForm[N[data1[[i]][[3]], 14], 14, 
     ExponentFunction -> (If[(# + 1) == 0, "", (# + 1)] &), 
     DigitBlock -> 3, 
     NumberSeparator -> "[ThickSpace][ThickSpace]"], i + 4, 
    data1[[i + 4]][[1]], data1[[i + 4]][[2]], 
    NumberForm[N[data1[[i + 4]][[3]], 14], 14, 
     ExponentFunction -> (If[(# + 1) == 0, "", (# + 1)] &), 
     DigitBlock -> 3, 
     NumberSeparator -> "[ThickSpace][ThickSpace]"]}, {i, 
    4}], {Item["i", Alignment -> Center], 
   Item["!(*SubscriptBox[(I), (i)])", Alignment -> Center], 
   Item["!(*SubscriptBox[(J), (i)])", Alignment -> Center], 
   Item["!(*SubscriptBox[(n), (i)])", Alignment -> Center], 
   Item["i", Alignment -> Center], 
   Item["!(*SubscriptBox[(I), (i)])", Alignment -> Center], 
   Item["!(*SubscriptBox[(J), (i)])", Alignment -> Center], 
   Item["!(*SubscriptBox[(n), (i)])", Alignment -> Center]}]

(* Display the data as a grid *)
tbl02 = Grid[tbl01, Alignment -> {{{Center, Right, Right, "."}}}, 
  Spacings -> {{0, 1, 1, 1, 3, 1, 1, 1}, Automatic}, 
  ItemStyle -> {Directive[FontFamily -> "Times", 
     FontSize -> 14], {Italic}}, 
  Dividers -> {None, {1 -> Directive[Thickness[1]], 2 -> True, 
     6 -> True}}]

(* Center between margins by displaying as a grid in a grid *)
Grid[{{tbl02}}, ItemSize -> Scaled[1.0]]

The result is