It is not mandatory to take GRE and TOEFL to do your MS or MB in the US. You can get an admission only with your TOEFL or IELTS score, however you should be mindful of the fact that the colleges that are willing you provide an admission with just a language test is at the bottom rung of the ladder. GRE is accepted by most colleges offering an MS. Both GRE and GMAT will be accepted by Most B-Schools so if you are looking for an admission in reputed school you should either write GRE or GMAT
The pertinent question is why should we take a language test as both GRE and GMAT themselves have English in the test structure. The answer is very simple, the sections that appear to be Testing English in Both GRE and GMAT are not just testing English rather it is testing the candidates verbal reasoning which is combination of reasoning and language. So if these tests do not test your skills in English language then what do they test? They test your verbal reasoning skills, quantitative reasoning skills and analytical writing skills. The Universities a test your reasoning skills because they believe that these skills are essential for being successful in graduate or post graduate programs. These standardized tests help do two things .Firstly scores help admission council compare applicants who come from a plethora of backgrounds. Secondly the scores in the standardized tests predict the potential of student to succeed in the graduate studies by evaluating their reasoning and analytical skills.
Tests like TOEFL, IELTS and PTE are taken by non-native speakers of English to prove their credentials in English Language. This is mandatory for every non-native speaker of English who wants to either migrate to an English speaking country or Pursue Higher studies in that county. TOEFL accepted primarily in The US for Higher studies. Australian immigration authorities also started accepting TOELF recently. However IELTS can give you access to countries such as Australia, Canada, The United Kingdom, The United States, New Zealand and even Singapore. The IELTS is used for both immigration and Higher studies.