Would every particle in the universe not have some form of measurement occurring at any given time?
What you describe is the process known as decoherence: any interaction of a quantum system with its environment (e.g. with photons or other particles passing by, and, yes, most likely interacting through gravity, although we don't have a theory to fully describe this yet) has the potential to destroy its genuinely quantum nature, turning quantum superpositions into mere classical statistical ones. This process is indeed the first half of a measurement, the second half being the reading out of the result which resolves the remaining statistical superposition into a single result.
But decoherence is not an all or nothing, instantaneous process: it is progressive in time, and the weaker the interaction between a system and its environment, the slower it will decohere. When we actually do a measurement we deliberately arrange for the interaction to be strong enough and we wait long enough for full decoherence to occur, so that a result can be obtained. But in between deliberate measurements, we can arrange for decoherence to be so weak as to be negligible, at least for the duration of the experiment, so that the evolution is (almost) truly quantum. It's relatively easy for, say, single atoms at very low temperature, but it becomes harder and harder the bigger the system is (it is for example a well-known and very real hurdle to design quantum computers with enough qubits). In practice, gravity is not usually the limiting factor here, because it is such a weak interaction.
This topic gave me trouble as well. The fundamental basis for answering it is to look at decoherence.
Basically, any interaction in quantum mechanics yields the expected coherent result that comes from two particles interacting. Often this leads to an entanglement of their states. If we had constructed the particles with a known previous state, we can make statements about the state of the particles (such as probabilistic statements about momentum or spin).
However, what if we don't know any information about one of the particles? What if it came in from the outside environment? As such, we have no knowledge of the state. The best we can do is talk about its state as a random variable and apply statistics. The result is a density function showing the probability that our particle under test is in any given state.
Do this enough times with particles whose state is Independent and Identically Distibuted (IID), and the "quantumness" of the particle starts to go away. As the number of interactions goes up, the central limit theorem starts to apply, and the variance in the predicted resulting state diminishes. Eventually, when the variance is low enough, we start to say the particle is "measured" and that it has a state that matches the expected value.
This is, of course, a relatively new viewpoint. The original use of measurement was in explaining how the unusual quantum world could interact with the "classical" world, and in particular with classical beings such as us human beings. This has lead to the famous interpretations of Quantum Mechanics. Decoherence is another way to explain this effect. Instead of offering the philosophically perfect measurement of one of the interpretations, it offers a statistical process whose limit is the same as the predicted results of the other interpretations.
Of interest may be the concept of weak measurements. Weak measurements are designed to provide some measurement while retaining most of the quantum coherence.
Not every interaction is measurement or collapses the wave function. When light reflects off a mirror, the phase information is preserved. As each single photon hits the mirror and scatters on an electron, the photon doesn't hit the mirror in just one point or interacts with just one electron. Instead each single photon hits the entire mirror and interacts with all electrons in the mirror. In other words, due to the uncertainty principle, the interaction is a superposition of interactions with every electron in the mirror. This uncertainty preserves the wave function of the photon from collapsing.
The same concept applies to other collective processes, including the photon's travel through space, whether flat or curved by gravity. If the photon is allowed to take any trajectory, then the photon takes all of them simultaneously with different probabilities and therefore acts like a wave. In this case the photon's trajectory through space is a superposition of all possible trajectories. Therefore gravity does not collapse the photon's wave function (at least while away from black holes).
Furthermore, certain particles have a low probability of interaction, e.g. neutrinos that can fly through the universe as through a virtually empty space. Also, the hypothetical particles of dark matter may not nteract at all other than via gravity while the gravity interactions almost always would be a collective process described above that would not collapse the wave function.
Science is about predicting practical results. Your question however seems rather hypothetical. Whether the answer is yes or no, there seems to be no practical difference either way. Finally, quantum mechanics alone does not describe the universe as a whole. This requires quantum gravity to view spacetime as a function rather than a set of independent variables and effectively make this world a projection. Thus your question cannot be fully answered until quantum gravity has been developed.