Only a couple huge problems with his solution.
He is forcing insurance companies to pay for the contraceptives, abortions, etc because the Catholic institutions won't (and shouldn't). Let's think about this for a minute... how is this any different from the HHS mandate?
Answer, it's not. It requires a PRIVATE ORGANIZATION to pay for insurance for an individual. Sure it doesn't interfere with the Insurance companies religious beliefs (in most cases, I'm sure if you looked hard enough you could find a Catholic insurance company that actually follows its beliefs) however, where in the Constitution does it say the federal government has the ability to force a private company or individual pay for services to another individual? It's not much different than the individual mandate part of Obamacare that is up for judicial review by the Supreme Court in May/June.
You can say, "Hey, they're a health insurance company, they provide this stuff anyway." You'd be right, but they don't provide it for free. Insurance is not for preventative care. Car insurance does not cover your oil change, new tires, wiper blades, etc. It is to cover conditions that occur after you purchase the insurance that are not preventative. Birth Control and other contraceptives are preventative care, they should not be included in health insurance costs. Abortions could technically be covered since pregnancy is a condition, a medical change to the woman's body. HOWEVER it still does not cover the fact that the insurance company is going to have to pay for it, out of their own pocket. In the end who is going to pay for that? You and me, and anyone else that has that particular insurance carrier.
Preventative medical care is what the individual takes part in to ensure they do not have to pay the insurance premiums because they got sick, or pregnant. Using a condom, taking birth control, not having sex, eating healthy, exercising, not smoking, etc. These are all preventative and not something that should have to be paid for by an insurance company.
In the end it all comes back to PERSONAL RESPONSIBILITY. Free Choice. As an individual looking for a job you have a choice to work for a particular employer or not. If contraceptives and abortions are something you consider a must for your health care plan, do not go work for a Catholic-run organization. Or better yet, if it's the best paying job (or only job) around, find an insurance company that will cover you for those things separately. You can also speak to your employer about opting out of the company health care plan so you don't receive the paycheck withdrawal. Yes, paying for your own insurance outside of a company plan will cost more, but either way, you are responsible for you. Until Obamacare your employer was not actually required to provide health care (those places generally paid their employees a higher salary to offset the lack of a health care option), which is another reason I can't stand the AHCA.
Addition: After thinking on this some, I realized the White House's "solution" is actually a pretty lame copout that doesn't actually change anything. Who do you think is paying the premium for the healthcare policy? The employer. So they'll still end up paying for the cost of the birth control and abortions through premium payments, even if it's not part of the actual bill, because the insurance companies overall costs will go up for everyone to make up the loss from the religious companies. Assuming of course that any insurance company is willing to cover an obvious religiously aligned employer.