Like a lot of other companies, Verizon wants as big a piece of the pie as they can possibly get. They buy other companies, especially their computers, and keep branching out into other areas such as internet and cable provision. If they have their way, Verizon will be like companies in the past we had to break up.
Verizon is not trying to create a monopoly. The company has been going after the large companies in the sector, but it do not have a monopoly. It competes against industry heavyweights such as AT&T, Comcast, and Time Warner Cable. It has not attempted to purchase any of these, nor would it be allowed.
If you look at the totality of the market, it is AT&T and Comcast who are gobbling everything up left and right. Verizon is slowing down its roll out of more advanced land line fiber optic. In some cases, they are selling some of it off. They are very heavily focused on their wireless business. They are making some strategic purchases to improve their media and content avenues, but they aren't exactly purchasing large companies. The small outlets Verizon has sought are easily dwarfed by the mega purchases by Comcast of NBC and by AT&T of DirecTV. You want to see companies going big for a monopoly, there you have it. If anything, Verizon has been effective in mostly developing a home-grown network with their last major acquisition being Worldcom 10 years ago.
There are many competing companies for Verizon, who have taken their own share of the market. For example, AT&T recently partnered with DirecTV, which is a vast market share. As for cellular coverage, AT&T and Verizon seem pretty evenly matched. The only way Verizon could create a monopoly is buying one of the other exisiting cellular providers, which I don't see happening.