Q:

Can we observe solar p-p fusion reaction somehow on Earth?

What are the properties of proton+proton fusion reaction
$p + p → 2H + e^+ + ν_e + 0.42 MeV$
making it hard to replicate on Earth? 
If we aim beam of protons to a can of water, won't we observe some number of these p-p fusions?
Do we have p-p fusion events on some well known accelerator like LHC or old ones?

A:

The reaction has been studied in accelerators

A bubble chamber study of proton-proton interactions at 4 GeV/c Part I—Elastic scattering, single-pion and deuteron production .Summary
Elastic scattering, single-pion and deuteron production have been investigated. The cross-section for elastic scattering is σelastic = (13.5±0.3) mb. The angular distribution has been fitted to dσ/d|t|=(dσ/d|t|)0 e −bt in the region of low values oft. The best fit givesb=(6.7±0.5) (GeV/c)−2 and (dσ/d|t|)0=(91±5) mb(GeV/c)−2. The cross-sections for ppπ0, pnπ+ reactions are respectively (2.6±0.3) mb and (9.7±0.4) mb. These reactions are dominated by the (3/2, 3/2) nucleonpion isobar production and by forward backward collimation of the nucleons. The production rates for the isobarsN∗++1238
,N∗+1238,N∗+1500 have been estimated, taking into account the experimental peripheral behaviour of the interaction. In the pnπ+ reaction they are (50±2)%; (10±3)%; (4±3)%. In the ppπ+ reaction the production ofN∗++1238 is estimated to be (45±10)%. The dπ+ and dπ+π+π- reaction cross-sections are respectively (0.03±0.01) mb, and (0.04±0.01) mb.

It is behind a paywall:Il Nuovo Cimento A (1971-1996),Volume 49, Issue 3 , pp 479-498
Deuterons are produced and there is enough energy left over to get a few pions in parallel. You may call it fusion, since one of the protons turns into a neutron, but, as observed in the comments, with the strong interaction.
So all it needs is enough energy so that the reaction can take place.
The difference with the sun is the energy available for the  interaction.  When the average temperature is at the energy where the reaction has a high probability it means a high flux of these interactions will happen spontaneously. The sun models use the weak interaction because at the core temperatures which are less than 15 million Kelvin  strong interactions are improbable. The chain of weak interactions is shown here.
You ask in the comments:

What namely cannot be observed in the lab? Transmutation of proton to neutron or pn fusion?

In page 41 here the transmutation of a proton to a neutron is discussed, one needs electron antineutrinos. I do not know of accelerator electron-antineutrino beams.

