That really depends on what you call necessary.
If you completely forget all about $SU(2)_L$ (say, in an alternate universe with no Weak Interactions). Then mass terms in the Lagrangian for quarks and leptons are not forbidden by any symmetry and you would not need the Higgs field to generate the mass of the quarks or of the electron.
Now, in OUR universe, all those particles interact weakly and then mass terms are forbidden and you need the Higgs to give masses to quarks and leptons.
The mass of the proton, on the other side, has very little to do with the Higgs. Consider that the light quarks that compose the proton have a pole mass of a few MeV (2 to 4 MeV), while the proton itself masses ~900 MeV. Where does the extra mass comes from?
The answer is binding energy. The quarks interact very strongly and the mass of the proton is the combination of the three real quarks plus a whole lot of virtual quarks and gluons. Even if we had no Higgs and the masses of all those guys were zero, the proton mass would not change much.
Since most of the visible mass in the universe is made of protons and neutrons, it is a bit of a fallacy to say that the Higgs field "is responsible for all mass". It is certainly responsible for the mass of elementary particles, but QCD is what generates "our mass". If anyone wants to check what would change if we did not have the Higgs, there is a good paper by Chris Quigg (http://arxiv.org/pdf/0901.3958v2.pdf), which also has a deeper discussion on mass generation by QCD.