The answer to the title is, "NO". God is not a vegetarian because HE DOESN'T EVEN NEED TO EAT! However, Jesus DID eat fish during His time on the earth, and multiplied fish for other peopel to eat, and even ate fish after His resurrection.
Luke 24:41-43 And while they yet believed not for joy, and wondered, he said unto them, Have ye here any meat? And they gave him a piece of a broiled fish, and of an honeycomb. And he took it, and did eat before them.
It is important for Christians to put human beings first. Some speak as if animals and humans have equal rights, but the Bible clearly says that humans are much more important than animals. Luke 12:24 Consider the ravens: for they neither sow nor reap; which neither have storehouse nor barn; and God feedeth them: how much more are ye better than the fowls?
Matt 10:31 Fear ye not therefore, ye are of more value than many sparrows.
Matt 6:26 Behold the fowls of the air: for they sow not, neither do they reap, nor gather into barns; yet your heavenly Father feedeth them. Are ye not much better than they?
Yes, animals are important and certainly should not be tortured just for fun. But animals and plants are to be freely killed and eaten, at least until the end of this age.
Gen 9:3 Every moving thing that liveth shall be meat for you; even as the green herb have I given you all things.
1 Tim 4:1-5 Now the Spirit speaketh expressly, that in the latter times some shall depart from the faith, giving heed to seducing spirits, and doctrines of devils; Speaking lies in hypocrisy; having their conscience seared with a hot iron; Forbidding to marry, and commanding to abstain from meats, which God hath created to be received with thanksgiving of them which believe and know the truth. For every creature of God is good, and nothing to be refused, if it be received with thanksgiving: For it is sanctified by the word of God and prayer.
As any honest reader can see, extreme teachings on animal rights and vegetarianism should actually be called the doctrines of demons, not only to be ignored, but to be silenced.