Why do you need ECDSA? Do you need a third party to be able to verify the message's authenticity, or do you need only the recipient to be able to verify the message's authenticity?
Suppose Alice is trying to send a message to Bob, keeping it secret from Eve and preventing forgery by Mallory. Suppose Alice and Bob can a priori share public information about each other from the telephone book. It is sufficient for Alice to use an authenticated cipher under the secret that she and Bob share via static/static Diffie–Hellman key agreement. This is exactly the application in Whit Diffie and Martin Hellman's seminal 1976 paper on public-key cryptography.
If Alice's private key is $a$ and her public key is $A = g^a$, and likewise Bob with $B = g^b$, where $g$ is a standard base (say $2$, in $\mathbb Z/p\mathbb Z$ where $p$ is the 2048-bit RFC 3526 group #14 modulus, or $x^{-1}(9)$, in Curve25519), then Alice and Bob share the secret $$h = A^b = (g^a)^b = g^{ab} = g^{ba} = (g^b)^a = B^a.$$ If they derive $k = H(h)$ and use $k$ as the secret key for an authenticated cipher like NaCl crypto_secretbox_xsalsa20poly1305, then they can exchange messages under that secret key.
As long as the key is secret to Alice and Bob, nobody else can learn information about the content of messages transmitted with the authenticated cipher. As long as the key is secret to Alice and Bob, nobody else can forge messages that are verified by the authenticated cipher. Only Alice and Bob can verify messages—which, for private the authenticity of messages, is all you need.
There may be more to the story: compromising the long-term key $a$ or $b$ would allow an adversary to retroactively decrypt all past messages; to prevent that we need a protocol that allows the parties to promptly erase their keys. (This is sometimes called ‘forward secrecy’, a glib term that sweeps under the rug exactly when or how promptly the keys are erased.) But ECIES with static keys, or static ECIES composed with ECDH, has the same vulnerability to compromise of long-term keys.