One of the worst things I see managers do is listen to bad PR people. These bad PR people come up with the idea that they need to rationalize something away. Their idea is that if you can come up with a reason why you did something stupid (no matter how silly the reason is), you can make your apology successful.
To which I reply: You aren't fooling anyone.
In fact, you are simply giving people more reason to not trust you - particularly the staff that works with you.
The most recent example of this comes from Este Lauder. An executive posted an inappropriate meme on his Instagram account. As you can imagine, it got a lot of backlash. The executive was suspended and had to apologize. In his apology, he included that he posted the meme before reading it.
If someone were to believe that, they would have to believe that it wasn't a lapse in judgement, but the sign of an ignorant and careless person. Most of us find it hard to believe that someone that bad could become an executive at a major corporation.
Which means we all tend to believe it isn't true. In other words, he isn't fooling anyone.
The best way to apologize - simply admit the horrible mistake. There is no reason for it that makes you look good, so just say "I was wrong". Feel free to add how you learned something and what you will do to be better in the future. Just stop coming up with reasons that none of us will believe.