I realize this question was asked a while ago, but I stumbled upon it when I was also looking for this same function. My solution is a little cumbersome and isn't a full solution to what you're looking for, but it combines both a tiny script and a little conditional formatting.
I first wrote a small script using the onEdit() function:
function onEdit(e) {
var ss = SpreadsheetApp.getActiveSpreadsheet();
var sheet = ss.getActiveSheet();
var targetCell = sheet.getRange('AD1');
var activeCell = e.range.getA1Notation();
targetCell.setValue(activeCell);
}
I chose 'AD1' as the target cell, as it was far out of the way and, if need be, I could also choose to hide that column.
Then I went over to the conditional highlighting and typed this in as a custom formula:
=ROW()=ROW(INDIRECT($AD$1))
Voila! Every time I edit a cell, it automatically highlights that entire row.
It's not exactly what you're looking for, as it won't automatically highlight the entire row as soon as you click on a cell ... only when you edit the cell. Also, if you have other formulas running and other conditional formatting going on, your spreadsheet can start to get slow. But this is the closest I've seen out there to a possible solution.
Much less as cool, but still somewhat functional regarding legibility is a basic highlighting of every other row. For example:
in conditional formatting: =ROW()=EVEN(ROW())