How do I make a StringGrid's columns fit the grid's width?

Use the ColWidths property, like so:

with StringGrid1 do
  ColWidths[1] := ClientWidth - ColWidths[0] - 2 * GridLineWidth;

And for a more robust and flexible solution, take all fixed columns into account and parameterize the column index:

procedure SetColumnFullWidth(Grid: TStringGrid; ACol: Integer);
var
  I: Integer;
  FixedWidth: Integer;
begin
  with Grid do
    if ACol >= FixedCols then
    begin
      FixedWidth := 0;
      for I := 0 to FixedCols - 1 do
        Inc(FixedWidth, ColWidths[I] + GridLineWidth);
      ColWidths[ACol] := ClientWidth - FixedWidth - GridLineWidth;
    end;
end;

procedure TForm1.Button1Click(Sender: TObject);
begin
  SetColumnFullWidth(StringGrid1, 4);
end;