Is to + ing (to becoming) correct?

Solution 1:

It would have to be to becoming a developed nation.

To clarify, the reason is that in this sentence, "to" is being used not as part of an infinitive (like "to talk," for example). It's a simple preposition; the sentence could be recast into something like "We're getting closer to the state of being a developed nation."

Solution 2:

The main issue here is that there are two main forms of the on track… phrase, one taking a noun (with for), one taking an infinitive:

We’re on track for our development goals next year.

We’re on track to reach our development goals next year.

The first of these, the for form, can be used with any noun phrase, so in particular works fine with a gerund:

We’re on track for reaching our development goals next year.

(‘Gerund’ means that reaching… here is acting as a noun phrase. Compare, for instance, “Reaching our development goals will be wonderful!”)

The main problem with the example you give is that becoming is a gerund, so becoming a developed county acts as a noun phrase, but on track to isn’t usually used with noun phrases — on track for is much more common. So you would not usually write or say either of:

?We’re on track to our development goals next year

?We’re on track to reaching our development goals next year.

In general, though, to is fine with a gerund, anywhere you could use to with a noun phrase:

We’re very close to reaching our development goals.

[There’s also a second, unrelated, much simpler issue in your sentence: it should be “becoming a developed nation.”]

Solution 3:

The expression "be on track + to ..." is more likely to appear in writings however the example given here (and very common form in conversation) implies a slightly different meaning: "We're on track to becoming a developed nation." means we are making progress and (The writer thinks and evidence shows) that we are going to start the path which leads to a developed nation "or" we are going to start becoming + a developed nation.