He'd be crucified... and then what? He owns the club.
Arsenal being the weakest club in the top 4? that's a reason not to sell. No replacements available? that's a reason not to sell. Suarez worth more than 40m? that's a reason not to sell.
Embarrassment and posturing? I don't really think that's a factor at all. The fans are powerless and likely pretty forgiving; they know they can't hurt Henry without hurting the club, and Henry (and every club owner out there) knows that too. If Liverpool sell, it'll be because it's a good business decision (or the best bad one) for the club and I don't think looking weak really matters. Liverpool said they wouldn't sell Torres, and now Torres plays for Chelsea; Wenger said he wouldn't sell both Nasri and Fabregas in the same summer, and then proceeded to sell them both.
A promise is meaningless.