No! It makes no sense. The idiom "crack the whip" refers to one's authority to get a job done. It is usually used in situations wherein the boss will have to "crack the whip" to finish the project on time. It means that the boss will have to push the employees to work faster and harder. One doesn't "crack the whip" on oneself. One can "get the lead out"; "knuckle down"; "get moving"; "apply some elbow grease", and so forth. "Crack the whip" can be traced back to 17th century horse-drawn wagons. The driver would "crack the whip" to get the horses moving forward. There's also a British expression that states: "a fair crack of the whip." It means to give someone a chance to succeed at something.

No. To crack the whip means to make other people work harder. I suppose you could say he doesn’t crack the whip on himself.

Once again, life is subjective. I agree with the literature! There is no definition in any of the texts, formal or urban, that define the idiom "crack the whip" as an action given to oneself. The usage is explained consistently as that of a superior or boss making someone else work harder. It also has the propensity to have a bad connotation related to slavery. I don't use it at all. I simply gave an opinion. And you know what they say about opinions? Of course you can reconfigure the idiom. You can do whatever you want. I am a native speaker of English and I can jumble around any expression, seriously or playfully. But, the normal, well-known idiom is not directed at oneself. If so, show me any works of literature that have done so, even colloquially.

I agree with Guidedogsaint, it would be best to say 'he doesn't crack the whip on himself.' This is a perfectly reasonable expression, we have many reflexive saying where our 'executive function' is forcing our 'lower self' to do something. "I'm forcing myself to pay attention,' 'I'm making myself do my homework,' 'I'm holding myself accountable for following this diet,' etc. There is also a long religious history of people 'self-flagellating,' or literary cracking the whip on themselves; this expression continues to have non-literal use, i.e. excessive self-criticism.

I would just add one thing to Terry's excellent answer: You can still hear "crack the whip" being used with its literal meaning today in horse racing. Jockeys often "crack the whip" on their horses to encourage them to run as fast as they can.