Yes, CEO's should be held accountable for remarks made in private. When you are a salaried employee, you are representing the company at all times and need to be aware of that. Any comments you make can be taken as comments made on behalf of the company and you should be held accountable.
It is absolutely a must that CEOs should behave themselves, even in private life, as CEOs. No shareholder wants to find out that the management of the company he has invested his savings has nude photos for example. The Facebook frenzy and the whole social webs show-offs have to be used extremely carefully by senior managers. Even the young and ambitions career hunters have to be really attentive with social media and private life extremes. So not to fear one day, at senior positions, of bad exposure.
I don't think CEOs should bee held accountable for remarks made in private. Everyone needs a little bit of private time. Even though CEOs are usually paid a lot of money, they should still only work for a set amount of time. No job should be a 24 hour thing.
What CEOs, or anyone says in private is a matter of their own personal opinion and therefore should not affect anyone or any company. Someone's opinions or actions usually do not have a direct affect on how they perform their job duties. If there is a conflict of interest then that can be dealt with.