"Committed to achieve"? (from Instagram bio of WHO)

I think that the answer is that at the moment, the passive form of commit followed by the infinitive form of a verb in not an established usage. There are examples of such a usage with other verbs, but not with the verb commit ... yet. So your sense of wrongness was justified.

Merriam Webster provides nothing in response to the entry "committed to", and under "committed" provides no parallel or example of "committed" with an infinitive. But it responds to "determined" as follows:-

Having reached a decision : firmly resolved.

And it gives the examples:-

was determined to become a pilot

is determined not to let it happen again

The Cambridge English dictionary set out its account differently, but re-enforces Merriam's account.

having promised to be involved in a plan of action:

It gives the following examples:

We are committed to withdrawing our troops by the end of the year.

The prime minister has said that the government is committed to the preservation of the country's national interests.

We are firmly committed to reducing unemployment.

In each example committed to is followed by a noun or noun phrase. That is why I say that, while there are usages that are analogous to the WHO use with an infinitive, so that it is obvious what they it is trying to say, it is not an established usage. I know I am disagreeing with the earlier answer which you have accepted, but I have to beg to differ.

Both are correct: they have different meanings. The infinitive form is what the WHO intended.

The infinitive form indicates the complete action and implies the completed action.

The "-ing" form indicates an action that, at the time referred to, is uncompleted.

The WHO's use indicates that they wish to attain that point at which there is "better health for everyone, everywhere", thus the "-ing" form indicates that they wish to attain a state in which the providing "better health for everyone, everywhere" has started but has not (at the time referred to) finished.

Have a look at the results and examples in Google NGrams for committed to achieve,committed to achieving