This works just fine until I reach a cell with no data, i.e., empty.
Is there a way to modify the code so it will exclude/ignore empty
cells and continue converting the remainder of the column?

Try this...

Dim X As Long
Dim ModifiedDateString As String
For X = 1 To Cells(Rows.Count, "A").End(xlUp).Row
If Len(Range("A" & CStr(X)).Value) 0 Then
ModifiedDateString = Replace(Range("A" & CStr(X)).Value, ":000", " ")
Range("A" & CStr(X)).Value = DateValue(ModifiedDateString) & " " & _
TimeValue(ModifiedDateString)
End If
Next

Rick