Do you agree with everything your boss believes? Is this a requirement for working under them? The challenge issued this week to Mozilla – creators of the Firefox web browser and advocates of the "open web" – is whether their new CEO's alleged support for anti-gay marriage campaigns is at odds with their inclusive, community-driven stance.

The boss in question is programming guru Brendan Eich, creator of the hugely popular JavaScript language and general web evangelist. Mozilla have been operating under an acting CEO, Jay Sullivan, for over a year and announced on Monday their decision to appoint Eich to the role after his 15 years there, most recently as CTO. The controversy stems from a $1,000 donation Eich made in 2008 in his own name – albeit with Mozilla's name listed alongside it, as US law stipulates – to the Proposition 8 campaign, an amendment to California law which outlawed same-sex marriage.

Mozilla is well known for its commitment to open and inclusive technology, pitching itself as a grassroots competitor to commercial companies such as Google. It didn't comment, though, when the controversy originally surfaced back in 2012, and Eich himself only acknowledged the apparent discrepancy in a blogpost which skirted around the issue.

Mozilla: "a global community who believe in openness and opportunity"

Today, things are more complex. The web is in a state of flux: on the one side, shadowy government agencies and enormous tech companies clamour to collect our personal data, with or without our approval. On the other side, Mozilla positions itself as "a global community of people who believe that openness, innovation and opportunity are key to the continued health of the internet". This ethos is clearly impacted by Eich's position, whether or not it is a personally held belief.

A CEO's political views and opinions shape that organisation's future: Mark Zuckerberg's burgeoning interest in US immigration law comes from a desire to widen the net Facebook casts in hiring talented people from around the globe. While it’s unlikely that Eich would suddenly introduce anti-gay attitudes to Mozilla's company manifesto, likewise it feels counterintuitive to the organisation's aims to let such things go unchecked.

The film studio Lionsgate found themselves in a similar position last year as their adaptation of Orson Scott Card's "Ender's Game" novel was met with controversy due to his homophobic statements. The studio eventually issued a statement in support of same-sex marriage and publicly distanced themselves from Scott Card's beliefs.

Such beliefs are, of course, the right of the individual, no matter how distasteful they may appear to others. This particular issue feels different, though, because of the nature of the belief. To actively oppose the union of two people who love each other – and to pitch in a significant amount of money to prevent that union being legally recognised – is a strong step in the direction of anti-equality. At least one prominent developer has already announced – with his husband – a personal boycott of Mozilla's products and tools.

The web is for everyone

An organisation should not be treated as a single homogeneous hivemind, with all employees sharing the same viewpoints. Diversity of belief and opinion is an important asset to a modern, innovative company. But issues of equality are more than just matters of conscience. These are the things that define us, our attitudes and our actions.

Eich's stance is unlikely to change. But by placing him in this leadership role, Mozilla ignores the unspoken message its silence implies: that it will turn a blind eye to a leader's presumed desire to shut down, invalidate and remove a whole subgroup's rights.

Nobody should have to to police their thoughts and tow the company line in their personal life. But to pretend that the opinions and actions of the CEO of a global organisation are independent of the way that company is perceived and the way its values are lived is to ignore an ugly truth. How can a company striving for openness appoint someone paying to oppose it?

The web community could live with Brendan Eich's opinion that same-sex marriage is wrong – that is his right to believe. But it can't live with his funding of this campaign which runs counter to what the web aims to address: that the internet affords us the opportunity to redress inequalities and barriers. As Tim Berners-Lee put it: "This is for everyone".

• Brendan Eich's appointment as Mozilla CEO causes ripples across company