No. He can't even PROPOSE laws by himself (in America). He has to ask the public to ask their congressman to submit it.
Presidents do not have to ask the public to ask their Congressmen to propose a law. Many times, the staff of the President works on proposed laws with or without input from Congressmen. Once the wording of the proposed law is in final form, the President works directly with the Congressional leaders in his own party and they work out which Congresmen will introduce the bill. Presidents do not ask the public at large to ask their Congressmen to propose laws.
Pursuant to Article II, Section 3 of the Constitution, the president may "...recommend...[to Congress]...such Measures as he shall judge necessary and expedient...". The president may, therefore, have a direct role in the legislative process by submitting bills written by him directly to Congress for their consideration.
The president "executes" laws by writing regulations pursuant to an existing statute. For instance, there is a Social Security Act passed by Congress. To implement the Act, the president writes rules and regulations that have the effect of a law. Regulations are published, comments from the public are accepted, and the relevant agency either orders the proposed regulation to be effective or not.
A bill passed by both houses of Congress does not become law until it is presented to the president for his approval and signature. If the president disapproves a bill, he returns it to Congress along with his written objections. This action is referred to as a "veto". Congress then reconsiders the bill and votes again. If two-thirds or more members of each house of Congress approve the bill it becomes law notwithstanding the president's veto. If a reconsidered bill receives fewer than a two-thirds favorable vote in either body, the bill fails. If the original bill sent to the president was passed by two-thirds of the members of the House and Senate it is considered "veto proof". It becomes a law whether or not it is approved by the president.
In recent years presidents have attempted, for all practical purposes, to invalidate portions of enacted legislation by issuing "signing statements" to Congress concurrently with his approval of a bill. These statements essentially announce that the president will not "execute" those portions of enacted legislation that he opposes. This practice is not supported by the Constitution, and is arguably in violation of it, since the president is obligated to "execute" all laws without reservation. However, the practice has not been challenged in the courts.