I consider the original sentence as more informal English than formal English, and is not strictly correct. The 'but' in this sentence is special in that it can technically split the sentence into two. 'He chases girls all day' is a phrase in itself in this situation, and is the actual meaning of the sentence. The sentence is NOT intended to say 'He does chase girls all day' , which 'does' was added as an emphasis that he really did chased girls all day (emphasis was added to express the author's view that chasing girls all day is equivalent to doing nothing. But the sentence's meaning is different if 'nothing but' is removed). Therefore, I consider it more correct to use 'chases' instead of 'chase'. The 'does' in this case can only be interpreted as a part of 'does-nothing' as a sentence, but not as a case of auxiliary verbs though it may be technically acceptable to say so.
I disagree the two sentences by themselves are contradictory. The first sentence was said for its rhetorical effect. ("He does nothing at all. He just chases girls every day." 'Just' was added here, but it DOES reflect the true meaning of the original sentence.)
For those who consider this an appropriate use of do-auxiliary, consider this. In all normal uses of do auxiliaries, the sentence will not be able to stand on its own without a change in meaning if the actual verb is removed. But this is not true in the sentence of interest here. 'He does nothing all day' is a sentence in itself. Also, if a do-auxiliary is indeed what is going on here, replacing 'nothing but' with 'except' / other than should produce the same sentence, but it is clear that it would not be correct unless a gerund is used. Notice that "He does nothing other than/except this all day" are both valid sentences as well.
Notice how all of Azor's example did not have the 'but' conjunction word in it. In essence, the examples are different from what you have in your question. "He does nothing all day" is a valid sentence with a meaning. Whereas "He didn't her", or "He usually doesn't call", or "She doesn't take her lunch break." are either not sentences , or are sentences with COMPLETELY different meanings (i.e. they cannot stand on their own). So while his examples are correct, it does not apply to the sentence of interest of this post.
The way you can argue for a bare infinitive in this sentence is to see it constructed as "He does nothing but (this) all day". "This" in this case is "chasing girls all day" / "chase girls all day" (noun phrase). However, you don't use a gerund with 'but' generally, so a bare infinitive is used. However, this is really only 'borderline correct' and is generally not a very good way to say this. It gives the sentence an air or informality, and is seldom found is formal writing.