Is the present pressure in the universe negative?

Is the present pressure in the universe negative?

Yes. In the current standard model of cosmology, known as Lambda-Cold Dark Matter, there are three contributions to the energy-momentum tensor $T^{munu}$ on the right side of Einstein's field equations: radiation, matter, and dark energy.

Radiation has pressure, but radiation was cosmologically significant only in the early universe, say the first 10 million years. This is because its energy density drops rapidly as the universe expands, as the inverse fourth power of the scale factor: $rho_text{rad}(t)propto a(t)^{-4}$.

Matter (by which cosmologists mean non-relativistic matter) is well-approximated as having zero pressure. Only highly-relativistic matter has significant pressure in cosmological terms, and then it behaves like radiation.

This leaves only the negative pressure of dark energy as the total pressure in the universe today. In the current theory, this pressure has the value

$$p_Lambda=-Omega_Lambdafrac{3H_0^2c^2}{8pi G}approx -5.4times 10^{-10}text{ Pa}.$$

Here $H_0$ is today's Hubble constant and $Omega_Lambda approx 0.69$ is the fraction of the current total density of the universe that is believed to be due to dark energy. This number comes from studies of the small anisotropies of the cosmic microwave background.

It seems the negative pressure is characteristic of any gravitational system.

No. It is not characteristic of gravitating matter or gravitating radiation. It is a peculiar property of dark energy, which does not behave like either matter or radiation. However, it is not too peculiar. For example, in quantum field theory, a scalar field can have negative pressure.