The dissociation of the COO-H bond forms a fairly stable COO^{-} anion, so it is acidic, however when compared to the stability of, say the conjugate base of HCl, the anion Cl^{-} has a full noble gas configuration so it is much more stable and the H-Cl is therefore much easier to dissociate.
Thermodynamically
HX \leftrightharpoons^{\deltaH} H^{+} + X^{-}
The enthalpy required to break this bond should be positive, I.E. energy must be supplied to break it.
H^{+} + H_{2}O \rightarrow^{\deltaH} H_{3}O^{+}
The formation of the hydronium ion in solution will have a negative enthalpy change, it will release energy.
A further interaction is also noted, the hydronium (solvated proton) ion will not exist on its own in solution and will be surrounded by water in order to stabilize the positive charge (make less positive):
H_3O^+ + nH_2O \leftrightharpoons^{\deltaH} H_{3}O^{+}.nH_{2}O
These combined effects, along with similar water based stabilization of the X^- ion will both have negavie enthalpies. And in terms of thermodynamics the energy put into split the HX should be outweighed by the energy gained from solvation and hydronium ion production. So in theory it would seem ok to suggest that all acids would just want to go the path of thermodynamics and become completely dissociated. However the reverse reaction has an effect.
The anion in solution, if not completely stable will readily protinate to reform the acid:
X^- + H_3O^+ \leftrightharpoons HX + H_2O
Although in the instantaneous addition of the acid to water (i.e. at t=0) the reaction for dissociation of the acid will be completely forward, as soon as the concentration of conjugate base starts to increase, the reverse reaction will start to take place. And so we reach an equilibrium, where the reverse reaction happens at the same rate as the forward reaction and we, the observers, see no change in pH. The reverse reaction for a strong acid is so thermodynamically weak that it barely takes place, weras for an extremely weak acid, such as ethanoic acid, it has quite noticable stopping power on the forward reaction.
As a side note, trying to get an "acidic" gas like HCl to dissociate into a H^{+} and Cl^- in space, with no water molecules around to bind, is thermodynamically unfavourable due to the production of a "Naked" proton rather then a "Solvated" hydronium ion. This is a quite common textbook argument for why chemists tend to write H_3O^+ rather than H^+ although it taken me about 2 weeks of organic synthesis lectures to get bored of the convention and stick to the easier (yet "Technically" innacurate) H^+ notation.
