Why does carboxylic acid not dissociate completely?

The bond between the oxygen and the hydrogen is a covalent bond, with a dipole. The oxygen tends to pull electron density off the carbon, but it can pull electron density off the next carbon, (and so on, if it's a long chain carboxylic acid)

So the electron density that it needs to pull from the hydrogen is quite small.

So the proton of the hydrogen atom isn't that exposed.

Compare that to the bond between hydrogen and chlorine.

Chlorine only has the hydrogen to pull electrons from, so while the electronegativity difference between the two atoms is less than between oxygen and chlorine, more electron density is lost by the H.

So when a water molecule comes along, the H prefers to sit on the lone pair of the oxygen atom in a water molecule, making H3O+, rather than hang around covalently bonded to a chlorine that is taking the lion's share of the electrons in the bond.

Back to the carboxylic acid; The proton can stay as part of the molecule, covalently bonded, and having a small share of the electrons, or it can let go of the molecule, and be adopted by a water molecule, where it'll have a lone, but be part of a positive ion; H3O+ If it does that, it'll probably have less electron density than if it stayed where it was, in the molecule.

So, in weak acids, most of the protons stay as part of the molecule.

In strong acids they get a 'better deal' by dissociating from their (ionic) partner (Cl-, NO3- for example), and giving up their own electron for a better share of the oxygen's lone pair.