I have a sharp LC-60LE600U and it will turn on for 1 second (gray screen) and right back off... After that it will refuse to come back on until unplugged. Any idea on what part that could be?

Yes most likely it is the power board. if you want to save money and not purchase the complete board, you can replace the capacitors on the board. Sonys are known for faulty capacitors. If they look swollen then they are bad.

if this tv is still under warranty i would be filling a claim straight away this is almost always wither a hardware or firmware fault both of which if stock should be covered under warranty.

good luck.

Fault to the main power supply regulator section. Either it will be a separate board, or the part of main board. Check and replace damaged component/s at its main power regulator [SMPS] section circuit. Contact any service technician. If you wish to get some details; check the site linked here. Pull up older posts.
http://electronicshelponline.blogspot.com/
It will be best to replace the power supply regulator board as card basis rather than making "Surgery" to it; which usually will make loss for both money and time. Even though you can make it right, it will cost around an amount equal to the cost of a new board; including the cost of (spare parts + service charge) More to say, most of the spare parts to it will be difficult to get from open electronic spare part market. This is true to most of the flat panel TV power supply boards; irrespective of its brand and screen size.

Sounds like corrupted firmware. A firmware update is necessary. Contact the authorized service center for Sharp.
If you wish to get some details; check the site linked here. Pull up older posts. Surf the site with patience.
http://electro-medical.blogspot.com/
http://electronicshelponline.blogspot.com/

If you bought it with a credit card that usually gives you a year of extended warranty so repairs would be no cost

On my older sharp non-smart TV there is a back/return butting on the remote.

The TV you have has a 1 year warrantee. Call 1-800-237-4277 (1-800-BE-SHARP) for customer support.