It's doable, but requires macros. I don't know how well GS can handle it. Many mobile editors seem to get stuck on macros.
Alternatively [ Ctrl ; ] is a nifty shortcut 
Or https://yourbusiness.azcentral.com/i...cel-10658.html
But if you want the macro route...
Code:
Option Explicit
Private Sub Worksheet_SelectionChange(ByVal Target As Range)
If Target.Cells.Count > 1 Then Exit Sub
Dim lookupRange As String: lookupRange = "A1:A10000"
Dim timeNow As Date
If Not Intersect(Target, Range(lookupRange)) Is Nothing Then
If Target(1, 1) = "" Then
timeNow = DateTime.Now
Target(1, 1).Value = Format(timeNow, "yyyy-MM-dd hh:mm:ss")
End If
End If
End Sub
Add the macro into the sheet...
It will only insert the date if the cell is empty.
I'd just leave the date/time as is and format the cells rather than modify the code to spit date OR time.
Change the cells range watching for click as needed...