For an infant, the nipple is not an intuitive interface. Children do not see a nipple and understand how to use it; it is a physical reflex called the Sucking or Rooting Reflex that is a series of motor motions an infant will perform if anything is near it's mouth. That's why infants will suck on your finger as readily as a nipple, and why they will suck on a nipple even when their eyes are closed.
Infants don't see the nipple and immediately understand how to use it; their brains have simply been hardwired to perform those actions. They perform other odd actions like "walking" their feet when they're held off the ground. You'd hardly call that "intuitive"; it's just stuff their bodies do.
Now, after a child begins to lose these primitive reflexes, they still "know" how to use a nipple. How? The knowledge has been reinforced. I see this round thing, I suck on it, I get yummy milk. The knowledge is even generalized to other, similar stimuli. It can take some work to do this; for example training a baby to use a bottle. But the association between suck -> food is still strong.
Now that's a pretty complicated learning process. Surely you can't say it's intuitive to suck on a nipple, right? Well, no, the nipple is still intuitive. The fact of the matter is learning and intuition are not mutually exclusive.
As beings with brains and minds, we learn. It is not always a conscious process; in fact it is almost always an unconscious process.
While this statement is pithy and cute, it fundamentally misinterprets both learning and intuition. I strongly encourage all of you to have more respect for psychology and the brain than to reduce it to pithy, inaccurate and harmful phrases like this. Learning is not "bad". Learning is what our brains do.
Let's get on track and understand Intuition as a psychological term. From Wikipedia:
In more-recent psychology, intuition can encompass the ability to know valid solutions to problems and decision making. For example, the recognition primed decision (RPD) model explains how people can make relatively fast decisions without having to compare options. Gary Klein found that under time pressure, high stakes, and changing parameters, experts used their base of experience to identify similar situations and intuitively choose feasible solutions.
Intuition isn't doing without learning; it's doing without thinking. You grab your mouse and you move it right; that's intuiton. If you had to sit there and think "Hm, I want this arrow to move right. How do I move this bar of soap to move this pointy arrow" then that wouldn't be intuitive because it would require rational, deliberate thought.
Sure, moving a mouse is not a "natural" interaction; how "natural" the reaction is is completely irrelevant to intuition. In practical terms, the only "natural" is what happens. For human beings in the United States, a keyboard and mouse is "natural". Natural in this context is the environment you're in, not the environment that would be if there were no humans.
Now, intuition is not always achievable as a goal either, so you shouldn't put undo focus on wanting to be intuitive. I can charitably read that remark as an appeal to not depend on "intuition", but as Michael Zuschlag's answer demonstrates, "intuitive" is also not completely impossible to achieve in an interface.
You know what's intuitive? Seeing a submit button and clicking it to submit a form. Sure, it's learned. Sure, stick a baby in front of a monitor and it won't fill out the form and "magically" know to press the submit button. But that's not what intuition is; intuition is understanding with only basic perception of an object.
For more on more natural learning, see my related answer on Cognitive Science explaining how learning is a very natural thing (even when it doesn't involve mice and keyboards).