The actual best way to "counter" someone criticising your tone is to try to avoid having them do so in the first place, by avoiding the problematic tone.
What's written below applies to after they've criticised your tone, but a lot of it (specifically the "rephrase" part) would also help with avoiding a tone that may offend others.
Step 1: Apologise
This may not be the advice you're looking for, and you may not believe you've done anything wrong (and this may be true). But it would still be the best way to deescalate the situation.
It doesn't have to be a particularly complex apology. Simply saying "sorry" and pausing for a few seconds could work well enough.
From here you have a couple of options:
Think what you're actually hoping to achieve in this discussion, how likely it is to happen (especially considering how reasonable you believe they are, and whether they even have the power to change the decision) and whether you have anything more to say or whether you'd just be going in circles.
If you're criticising a decision, but you don't have a good alternative, or you're simply being told about a decision that's already been made, there may be little that can be gained from trying to change their mind.
Some points I've made below may also lead you to dropping it.
Delay or move the discussion
This is probably the best way to continue the discussion if you really need to (which may not be true) and you're unable to rephrase what you've already said in a "better" way.
You could say something along the lines of "Can we continue this discussion tomorrow? I need to spend some time thinking about it and getting my thoughts in order".
You could, as another answer suggests, move the discussion to email instead. Or you could invite some other people to join the discussion (although be cautious with this, as it may be seen as an attempt to embarrass them or undermine their authority).
Rephrase what you've said, or take a different approach
Following on from step 1 with "what I meant to say was" could be a good way to transition.
This is probably the most difficult option if you don't know what the problem with your tone was. It's also difficult to give specific advice for - it would be much easier to answer this for a specific scenario. However, there are a couple of things that it might help to keep in mind:
How you said something is often more important than what you say
The world may be simpler for some of us if everyone were logical and could look at the facts presented without being swayed by how they were presented. The world unfortunately just doesn't work like this.
How you say things is very important.
I feel this is especially relevant considering you say you used a logical argument with numbers and facts, yet you say nothing about the tone with which you said this.
Tone is subjective
You may believe your tone is fine. If other people have a problem with it, they're not any more right than you are. But then you would be faced with the choice to either try to improve your tone or just live with the fact that interacting with those people will be difficult.
It's not "you versus them"
You're not trying to (or shouldn't be trying to) "win" the argument. You're trying to help them see the error in their ways, or work together to find the solution that's best for the company, or whatever else.
Compromise is good
Some arguments are simply not worth having (or continuing). In some cases you may not be able to convince them, in other cases the difference between the end results would be negligible when looking at the bigger picture.
It doesn't matter who's "right" if you're just wasting time.
They are your manager
They are the decision maker. If they want to make a terrible decision, you can, and should, try to guide them in another direction, but ultimately it is their decision to make. Even if this ends up ruining the company, or they end up blaming you, it's still their decision to make, and it's still not your place to try to stop them after they've made up their mind. Well, you could try to go above their head, but that's a whole other question (and generally won't go well). Or you could decide to find another job if their decisions are that bad.
The above may not apply to exactly as is to every situation, but understanding your place and keeping that in mind should put the discussion into proper context. It's more you giving them some information or perspective they might be missing, and less a discussion among equals where they need to defend their point (actually every discussion, regardless of with whom, is likely to go better if you approach it from the former point of view instead of the latter).
Dismissing is bad
You do not ever want to say something along the lines of "that's a terrible idea" or "you're wrong". It's generally more constructive to just remove these statements and, if applicable, just stick to the part where you justify this instead.
Questioning is better
If you think something is a bad idea, you can ask questions to lead them to reach the same conclusion, or their answers could reveal some information which changes your mind instead.
A simple "have you considered using X instead" would be much more productive than "using X would be so much better".
Avoid too strong statements
If they might have some personal investment in what you're directly or indirectly criticising, it might be best to downplay it. For example, instead of "users absolutely hated it", you could say "users were not at all fond of it". You could even ease up on that more by dropping the "at all" (whether this makes sense heavily depends on the message you're trying to send and how core this is to your argument).
Note: There may be people who "don't like your tone" simply because you question or disagree with them in any way, shape or form. In this case you'd strongly want to tend towards dropping it, and just avoiding questioning or disagreeing with them wherever possible (while also looking for another job). Although in my experience these people are very much in the minority.