The short answer is no, he probably wouldn't get very far just trying to do that! He'll need some hugs eventually.
To explain it the long way, hand to hand combat is just one of many examples of a situation where people would need to come into physical contact with each other, just like a handshake. However, because instances of physical contact while fighting are fairly brief, trying to do that and nothing else will just hold off the inevitable. Unless Levi was literally doing nothing but fighting 24/7, he would eventually start to feel the symptoms of touch withdrawal, just at a slower rate than someone getting no physical contact at all, and it would catch up to him. More prolonged physical contact (hugs really are ideal!) is necessary to hold off symptoms entirely.
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To explain it the long way, hand to hand combat is just one of many examples of a situation where people would need to come into physical contact with each other, just like a handshake. However, because instances of physical contact while fighting are fairly brief, trying to do that and nothing else will just hold off the inevitable. Unless Levi was literally doing nothing but fighting 24/7, he would eventually start to feel the symptoms of touch withdrawal, just at a slower rate than someone getting no physical contact at all, and it would catch up to him. More prolonged physical contact (hugs really are ideal!) is necessary to hold off symptoms entirely.