Splitting round trip tickets for one way trip

Yes, they will cancel the other leg if you don't show up for the first one.

You can't throw away the 1st half - they will cancel your whole itinerary if you do that. You might (emphasize "might") be ok to cancel the 2nd half after you travel, if you booked it the other direction. Maybe someone else can address whether you cancel or just be a "no-show" for the 2nd half - you will not get any money back, though - at least I wouldn't think so for a cheap fare. However, I've always been concerned that the next time you try to buy a ticket from that airline, there may be a block or something (I don't know that, I've just wondered if it would happen). I assume you'll have some proof of transit (another way to leave Italy)? I don't know that Europe countries are really strict, but some places are not comfortable with one way tickets.

Yes - they will cancel the rest of your ticket when you don't show, which is the flight you need. What about booking round-trip Tel Aviv -> Rome -> Tel Aviv? That way when you don't show for the return flight to Tel Aviv it will likely not matter unless they keep the record should you ever book with them again.
Buthave you looked at Skyscanner.com? If you are intent on doing this, it shows round trips beteween these two for in the $300s, and it shows a 1 way direct flight from Tel Aviv to Rome (FCO) for $232 (on Israir).

I can onlhy speak about Delta, but if you book a round trip and only use the first half, and just don't show up for it, the return part of the ticket is still usable at another date within one year. You just have to pay a change fee as well as any difference in fare. But, the others are correct that if you do not show up for the first part, the whole ticket is cancelled. However, the same applies as above, in that it is still usable with the additional fees, although no money back.

Thanks for the replies. After I posted I did think about doing a TLV-FCO round trip so I'd be throwing away the second half, not the first. I'll keep checking your responses about this option. I also found the Israir rates but don't seem to be able to actually book these flights (July travel). I haven't figured out why yet. Any thoughts or explanations?

Don't know why you can't get there, I just got to the Israir page to purchase Tel Aviv to FCO on July 9 for $292.....

The other issue is that in the bowels of the fine print may be a clause that allows them to bill you the difference between a one way ticket and any discounted price you may have paid if you fail to fulfill the terms of purchase. Read the conditions carefully.

Problem solved. Israir must have been doing website maintenance when I tried to get a ticket. It looks like I'll be able to get a one-way through them now.