Is RSA in a ECB-like-mode safe for bulk encryption? Let's say I would like to communicate with my friend using asymmetric/public-key encryption, e.g. RSA.

(Note: I do realize that in practice this is done through an intermediate symmetric key, but this question assumes we only use asymmetric encryption.)

Say I try to do this: I slice up my data into 2n-byte blocks (padding with zeros [edit: padding appropriately] if necessary), append the block index to the block (to prevent the same plaintext from turning into the same ciphertext), then use my friend's public key to encrypt each block separately.

Is this a secure scheme, or does the fact that I'm re-using the key make it susceptible to some kind of attack? If so, is it used in practice in any existing algorithms? If not, is there any way to make it safe, aside from using a symmetric key?
Well, reusing a key isn't a problem; after all, RSA keys are generally used many times.

However, if you fix the padding, there does exist one other potential problem; message malleability.

To example, suppose Alice sends two messages to Bob, $X_1, X_2$ and $Y_1, Y_2$.  To send these, Alice actually sends:

$E(X_1), E(X_2)$

$E(Y_1), E(Y_2)$

Now, Eve can't modify each individual block (RSA with proper padding prevents that); what she can do is mix-and-match blocks, as so:

$E(X_1), E(Y_2)$

This would decrypt at $X_1, Y_2$, which might not be what Alice and Bob whats.

If you are silly enough to use RSA encryption in this manner, you do need something that binds the blocks together (or alternatively does some sort of integrity checking on the entire message).

As for whether people ever do this in practice, well, no they don't -- not because of security, but because it's so inefficient; using the RSA algorithm to communicate a symmetric key, and using that symmetric key to encrypt the message is far more efficient.