Is the water-splitting enzyme a component of photosystem II?
The name of the protein (actually, three proteins) you are referring to is Oxygen-evolving Complex.

And yes, it is part of Photosystem II.

According to Lodish (2002):


  An Oxygen-Evolving Complex in PSII Regenerates P680: The splitting of H2O, which provides the electrons for reduction of P680+, is catalyzed by a three-protein complex, the oxygen-evolving complex, located on the luminal surface of the thylakoid membrane. The oxygen-evolving complex contains four manganese (Mn) ions as well as bound Cl- and Ca2+ ions (see Figure 16-44); this is one of the very few cases in which Mn plays a role in a biological system.


You can see the complex in this diagram of PSII, also from Lodish:



Figure 16-44: In a reaction center, two integral proteins, D1 and D2, bind the special-pair chlorophylls (P680), two other chlorophylls (Chl), two pheophytins (Pheo), one Fe atom, and two quinones (QA and QB). All of these are used for electron transport following light absorption by an associated lightharvesting complex (see Figure 16-42). Three extrinsic proteins (33, 23, and 17 kDa) comprise the oxygen-evolving complex; they bind the four Mn2+ ions and the Ca2+ and Cl− ions that function in the splitting of H2O, and they maintain the environment essential for high rates of O2 evolution.

As you can see, all proteins coloured green in the image above are part of PSII. However, since the oxygen-evolving complex is made of extrinsic proteins, it can be easily separated from the rest of PSII:


  The Mn ions together with the three extrinsic proteins can be removed from the reaction center by treatment with solutions of concentrated salts; this abolishes O2 formation but does not affect light absorption or the initial stages of electron transport.


Source: 


Lodish, H. (2002). Molecular cell biology. 4th ed. New York, N.Y.: Freeman.