Can I sue my doctor for not treating an weeping wound immediately?

First, what would you be suing for--one week of pain? The out-of-pocket cost of the ER visit (since you can only recover the out-of-pocket cost, not anything paid or reimbursed by insurance)? Hopefully, you have not suffered any serious, long-lasting injury or disability due to the delay, or thousands of dollars of medical costs because of it; but if you have not, there is no point in suing. You can only recover in a malpractice suit the additional out-of-pocket costs (and lost wages, if any) caused by the alleged malpractice (so the extra costs or losses; not what you would have incurred no matter what for the condition) and, for disability or life impairment lasting generally many weeks, or months, or longer, some amount of "pain and suffering." But to bring a malpractice suit, even if you were to do it "pro se" (as your own attorney--which is not recommended, of course), you would need at least one medical expert (a doctor) who would examine you, write a report, and testify if necessary. Such experts costs several hundred, if not a few thousand, dollars, if all you had was a week or so of additional pain and maybe another $200 - $400 in medical cost, you'd spend far more on the lawsuit than you'd get back.
Second, you'd have to show that the doctor was negligent or careless in his recommendation, not merely that he was wrong. The law does not require doctors to be perfect; sometimes doctors do everything right and it does not work out for the patient. A doctor is only liable if his medical care was unreasonably careless (negligent) or otherwise not within then-accepted medical care standards. It is not a given that this doctor's care was negligent: sometimes with a wound, the best advice is to leave it alone for a few days or a week and see how things develop. So the doctor may not have committed malpractice.