Does every object in the Universe have gravity? Space has no gravity, why?

In the intergalactic medium — the most dilute regions of the Universe between the galaxies — as CipherBot writes you'll find roughly one hydrogen per cubic meter, i.e. the density is $sim10^{-6},mathrm{atoms},mathrm{cm}^{-3}$, or $sim10^{-30},mathrm{g},mathrm{cm}^{-3}$, or, in terms of energy (since mass and energy are equivalent through $E=mc^2$), $sim10^{-9},mathrm{erg}^dagger$. In addition to this, you'll find 5–6 times as much dark matter.

Space itself can't really be considered "an object". Nevertheless, even ignoring the normal and dark matter, space does have energy: the so-called dark energy. We don't really know much about it, but we can measure its presence through its effect on the expansion of the Universe. But whereas normal and dark matter decelerates this expansion, dark energy has the opposite effect of accelerating the expansion. And since the energy density of dark energy is more than twice that of the other two components together, it actually dominates the dynamics of the Universe.

So, in this regard you can say that space does have gravity, although it's a "negative gravity". This phrasing is a bit misinterpreted, though. The dark energy is thought to be a negative pressure. Pressure has a energy density associated with is, so negative pressure has a negative energy density.

$^dagger$_{$1,mathrm{erg}equiv10^{-7},mathrm{Joule}$.}