How is message length involved in public/private-key encryption? I've just read this question regarding why to sign only the hash of a message. I kind of understood the answer, but the question arisen is now:

Why can I encrypt any text messages with the same method while it is needed to split up the message in order to sign it?

I mean, signing is taking the private key and encrypting e.g. message A and on the other side taking the encrypted message and using the public key to decrypt it. What is the difference that makes the other way around (public key for encryption, private key for decryption) (more) secure?
First let me emphasis again that signatures are something entirely different than public key encryption. Please read the comments to your question and keep those in mind. 

That said, in both cases (signature &amp; public key encryption) schemes are designed for fixed message sizes. For RSA signatures and encryption the maximum message length is determined by the size of $N = pq$ (assuming the school book notation for RSA). You can neither sign longer messages nor encrypt them. 

What applications will do is using a cryptographic hash function in case of signatures to reduce the message length to a fixed small value. In case of encryption they will use hybrid encryption, i.e. the application will generate a random secret key for symmetric encryption, encrypt it using the public key scheme and encrypt the message using symmetric encryption with the generated secret key.