I was able to get this to work by doing the following:

Created a macro called AutoFitCell

Sub AutoFitCell()

'Autofits the cell based on the information it is receiving

Worksheets("Executive Summary").Range("H:H").EntireRow.AutoFit

End Sub
Then created another Macro that activates the above code via a BeforeSave:

Private Sub Workbook_BeforeSave(ByVal SaveAsUI As Boolean, Cancel As Boolean)

'To run a macro named AutoFitCell
Call AutoFitCell

End Sub
Seems to be working.