Can I ship items home from the cruise line?

Some shops will ship; others will not. I'd guess that the higher-end shops would be more likely to offer that service. I don't know about cruise ships.

Be aware that shipping costs from Europe are very, very high, and if you don't handle the packaging-up and mailing yourself, the cost of those services is very likely to be passed on to you, making the process even more expensive. It might well be cheaper to pay the excess-baggage charge for an extra suitcase on the flight home so all your carry-on capacity can be used for breakable purchases.

Ceramic dishes, shipped from Italy. What could possibly go wrong here?

If it were me, I'd carry anything breakable in my carry-on luggage going home (and carefully). If you need more space, ship home something that you would have had in your carry-on, that's not fragile (and that you won't need for weeks).

Prepare for some sticker-shock when shipping things.

or will the cruise line offer shipping?

This tidbit puzzles me. If you are touring Italy, how does a cruise line factor in to shipping items?

The cruise companies won’t offer any kind of packing and shipping services. Some shops will ship, but charges are astronomical. Hand carrying your treasures is the best way to get the stuff home. Are you doing a TA? Just curious. US to Europe cruises go only in the spring. TA westbound is only in the fall (unless you’re on Cunard, but that’s out of Southampton).

We had some ceramics shipped from Italy some years ago. The shop handled all the packing and shipping, and everything arrived intact. I don't remember what it cost; it's been a long time. But I do know we were less able to afford things then, so I suspect the price was not exorbitant.