Can anyone offer some advice on how to approach an ethical complaint against leadership from one of their direct reports?

The compliant, like any other, needs to be investigated, a conclusion drawn, and appropriate action taken based on the findings of the investigation. That doesn't mean your leader did anything inappropriate. You just need to go through the process to determine whether the complaint has merit. Given this is a member of leadership, who should lead the investigation and ensuring no retaliatory action afterwards are more sensitive than usual. 

I am not sure of the size and structure of your organization, but in my experience non-employment ethics issues (e.g., financial misconduct, misuse of company resources, etc.) were investigated by legal/audit, or if truly needed, by engaging an external third party. Employment related items (e.g., harassment, discrimination, and other inappropriate workplace conduct) were investigated by HR. If there was concern that the HRBP was too close to the leader to be objective, we may bring in another HR person who does not have ties to the matter. 

I appreciate the politics could be a lot, but when done right this is a great opportunity to show your organization is committed to ensuring ethical conduct and will not shy away even if the concerns are brought up regarding a member of leadership. 

If this is about the CEO, the Board Chair will likely need to be engaged. Your legal counsel could help you think through how and when to do that. 

You are likely to get many perspectives to this question, but here is mine based on my experience.  This issue has levels to it.   First, when you talk about "Ethical" it depends on if the alleged behavior has an illegal/major inappropriate or minor inappropriate nature. If it is the former, you have to make everyone who needs to be in the loop aware, so an official investigation can occur because there could be legal implications and consequences.  If it's the latter, you might consider learning a little more about the situation to determine course of action which could be informal or formal coaching, education for the accused or the complainant, or some other remedial action.  It is always good to remember that these type of situations are always stressful and painful for all parties, so you want to employ the most discreet but appropriate  process as possible.  Good luck. 

This is so contextual it's difficult to comment really.  Are you in a business that demonstrably wants to be ethical?  Is this a leader who is considered sacrosanct by the people above him?  Do you have established procedures where annonymised complaints are evidently reviewed and acted on?  etc etc

It all depends on the risk profile inside the business of dealing with something like this.  

If it's high risk to do something about it you need to tread carefully and try to get some information from the complainant without raising the profile too high and probably talk to the leaders boss first.  In this environment there's unlikely to be an open process, you may cause trouble for the complainant that they don't want, and you've got to decide I guess whether it's something you're going to go to bat on.

If it's expected that such things be addressed promptly and properly, well then it's easy - you follow a standard investigation approach and take action depending on what you find.

Hi, we have an 'Alert channel' where people can file a complaint. This is then discussed in a specific team and investigators are appointed to discuss with both parties involved while keeping the identity of the direct report anonymous of course (no retribution risk...) In some cases we involve our external lawyer for advice but we do need evidence on what has been really happening to avoid a witch hunt in case people simply don't get along. It is a well defined and transparent process so employees know they can contact the Alert Channel and they are in good hands.