This particular quote of Hale represents his fundamental belief in what he does as representing "God's work." It also reflects the fundamental flaw in his character. Hale is convinced that if there "be proof so immaculate," he finds it acceptable to take a life. Where his thinking is flawed, and something that comes out later in the Third Act, is that death is something irreversible. Signing the "72 death warrants" that he already has, he begins to recognize that what the court is accepting as truth and fact, as incontrovertible evidence, might not meet his standard of "proof so immaculate." When Miller writes in the Act I stage directions that the citizens of Salem held a lighted candle that served to represent a beacon for all, Hale embodies some of these qualities with this belief and unquestionable faith in the "proof so immaculate." The quote means that Hale has no problem executing his duty if this basis of evidence is strong enough to compel him to do so.
Yet, it is within these lines that Hale's character is seen. Hale is wondering if the evidence given by the girls meets this idea of "proof so immaculate." In the second half of the statement, the idea of not having "qualms" is raised if the evidence accepted is not as strong as one could believe. When Hale speaks these words to Danforth, it is clear that he is "drawing a line in the sand," indicating to the judge that he has no problem doing what he must do, provided that the evidence collected represents this. It is here where Hale begins to start showing some of the first real moments of Hale's doubt in what is happening in Salem.