Chapter 9, which is properly titled "The Leech" describes Roger Chillingworth as just that: some animal force that has opted to attach itself to Arthur Dimmesdale in order to extract whatever life is left in the man, as this is the only way that he would get to find out what is the ailment of the soul that is driving the reverend crazy. Granted, leech was also a colonial name given to doctors (as one of their premiere methods for curing people was bloodletting through leeches).
By saying that Chillingworth was "scrutinizing" his patient, Hawthorne uses a play on words. First, Chillingworth scrutinizes the way that a doctor would, but he really is not a physician. He is a scholar who knows more than the average individual about many things, including the medical sciences. He infiltrates into the village and is accepted immediately at face value, which says a lot about the villagers' critical thinking skills. It is true, however, that later on he finally is seen for the impostor that he may be, and he loses the trust of the people since he starts to appear suspicious to them.
The second irony is that, at this point, Chillingworth's scrutiny is not of the medical kind, but of the personal kind. He is adamant in knowing what is eating away at Arthur. He makes more questions about the nature of Arthur's soul than at the state of his body, which is what he really should be asking Dimmesdale about. Chillingworth goes as far as telling Dimmesdale that there is a doctor of the soul to which he should confess his issues, which Dimmesdale completely refuses to do.
"Many, many a poor soul hath given its confidence to me And ever, after such an outpouring, Oh, what a relief have I witnessed in those sinful brethren, even as in one who at last draws free air, after long stifling with his own polluted breath.
It is all a cheap attempt to get questions answered that Arthur is just not ready to give up yet.
Therefore, the irony lies at the heart of what Chillingworth aims to do, which is to make Arthur confess. To do this, Roger poses as a friend and support system to the poor Reverend, grilling him with questions in the process and, perhaps, making the man more ill than what he was before.