Can an RSA encrypted message be altered?
Close; what would work would be if Eve encrypts 2 with the public key (forming $2^e \bmod N$), and then multiplies the ciphertext ($P^e \bmod N$ with that.  When Alice decrypts it (by raising it to the power of $d$, she'll compute:

$$(2^e \times P^e)^d = 2 \times P$$

which is what Eve wants.

On the other hand, this is assuming that Alice uses raw RSA, without padding.  When you add in padding (which any sane use of RSA encryption should use), this doesn't work.