Unfortunately there is only one answer here
Business is business is business
Let's see the situation from the market's point of view. Bob works for Acme who has a contract with Initech. Initech does not agre with Bob's personal issues, could they be 1) political ideas, 2) religious faith, 3) affiliation to labour associations, 4) health status or 5) sexual habits/orientation. I chose the above 5 to match my country's privacy regulator for the definition of "hypersensitive personal information".
Initech is basically bullying Acme for letting Bob out or more in general to pressure behaviour against Bob, who for instance is not involved in any criminal activities. This is also called lobbying, under some circumstances.
Unless Initech is subject to a religious regulation (since few religious laws exists around the world) so that not only companies must comply with ethical/moral rules dictated by law, but must also demand their customers/suppliers chain to comply with same rules, Initech is simply attempting to show force against Acme to take economic advantage. Porter depicts both the customer's and the supplier's contract force among the five.
There are two edge cases:
- Acme is a small company with Initech being their biggest customer
- Acme is a large enterprise with Initech being only one small customer like any
The OP's concern suggests the first, or a middle case which is between these extreme bounds. In fact, any company in the second edge case is a suicide customer if they can't replace their supplier easily, and surely won't do any damage to Acme. Actually trying to bull someone bigger than you only result in additional damage to yourself.
So I assume that the customer has a lot of economical power over Acme. Acme, which to remind is Bob's employer, has very few options in this case. This specific case should be escalated to the top as soon as possible keeping in mind that we are talking about an individual's own future and his right to freedom of thought. Top management must act to define a company policy valid for the future and applied since this very first case.
Firing the employee will likely result in legal consequences, and there is the risk that people supporting Bob's ideas will spread the word about Acme being discriminatory, adding reputation consequences over the existing. This is especially true if Acme works in the B2C market and risks more reputational damage than B2B companies.
Standing on Bob's side Defending Bob's right of speech is a civil duty, but while may or may not increase Acme's reputation, it will surely cause an economic damage and a stress for the workforce. Bob risks to be "marked" by coworkers as who caused a big customer loss.
Unfortunately, this is a lesson to learn for everyone. Market out there is aggressive, sometimes criminal. Who thinks that only mafias come with a knife to the neck of an enterpreneur is wrong. Political affairs are everywhere. Evil customers and suppliers seeking the worst excuse to act power on their counterparts are behind the corner.
There is no simple answer. I gave an answer but don't claim the accepted answer flag, because there is none. All answers in this thread are objectionable, including both the downvoted answer that suggests to let Bob go (by Mehrdad) or the one to support Bob and confront Initech (by Ed Heal).
I also wanted to highlight that social media policies are widely used by companies but have a different scope. They normally affect the ability for an employee to present himself/herself as an employee of Acme by either prohibiting such advertising or by requiring that any statement must be approved by the company first. Normally bloggers would use a disclaimer like "The opinions and information provided in this site are not to be intended as the official opinion of the Compant nor they are approved/reviewed by it".
It is unclear whether Initech simply wants not to deal with Bob (e.g. as sales representative, consultant) or really want Bob to be let go. The second smells really unbelievable to my eyes, as I never seen even the most ultra-Catholic company based in Vatican City complain for a supplier representative that is either homosexual or in a de-facto family (namely has children without getting married).
Normally these companies will only request to replace their representative, in which case it could be a deal to move Bob to another customer by explaining the situation and letting him post any licit content on social media. In this case:
- Acme does not lose business
- Acme can cover up the entire stuff easily
- Acme shows weakness to Initech, which in the future may go against them
- Bob keeps his work and possibly respect from workforce