Ivana gives a good general answer. Yes, I agree. I think you could make a convincing argument for at least two potential titles for Shakespeare: Father of Modern Literature and/or Father of Modern English Literature. The second of these is probably the one most cited, and the one that I believe is the most supportable.
Here's what Enotes has to say about Shakespeare's influence on World Literature:
Few dramatists can lay claim to the universal reputation achieved by William Shakespeare. His plays have been translated into many languages and performed on amateur and professional stages throughout the world. Radio, television, and film versions of the plays in English, German, Russian, French, and Japanese have been heard and seen by millions of people. . . . Novelists and dramatists such as Charles Dickens, Bertolt Brecht, William Faulkner, and Tom Stoppard, inspired by Shakespeare’s plots, characters, and poetry, have composed works.
The above excerpt from the article "William Shakespeare: The Dramatist," certainly provides support for the effect that Shakespeare has had worldwide.
More, however, than his contributions to World Literature are his indelible contributions to the actual English language itself. He invented over 3000 words, nearly half of which we still use today, not to mention the famous phrases, coined by Shakespeare which are a staple of our English language today.
Here are a few examples:
accommodation, aerial, amazement, apostrophe, assassination, auspicious, baseless, bloody, bump, countless, courtship, critic, critical, dishearten, dislocate, dwindle, eventful, exposure, fitful, frugal, generous, gloomy, gnarled, hurry, impartial, inauspicious, indistinguishable, invulnerable, lapse, laughable, and lonely
And this list only goes up to "L." Just imagine what our English language would be without the word "lonely," to single out one of the above list. Almost impossible to conceive, right?
Here are some of the phrases he coined:
all that glitters isn't gold; barefaced; be all and end all; break the ice; breathe one's last; brevity is the soul of wit; catch a cold; clothes make the man; disgraceful conduct; dog will have his day; eat out of house and home; elbowroom; fair play; fancy-free; foregone conclusion; frailty, thy name is woman; give the devil his due; green eyed monster; heart of gold; heartsick; hot-blooded; housekeeping; it smells to heaven; it's Greek to me; and lackluster
Again, the list only goes to "L." Just imagine not having, at your fingertips, the phrase "to catch a cold," and just think how often (especially through the course of the winter months) you use this phrase! There is no doubt that our Modern English language and English Literature would not be what they are today, if we took away the contributions Shakespeare has made.
So, though Shakespeare had a great influence on modern authors and literature around the world, and could be considered the Father of Modern Literature, I would have to say that his real claim to fame is the addition he made to the Modern English language that, in turn, had an indelible effect upon all authors writing in English who followed him. This fact alone would make a strong case for calling him The Father of Modern English Literature.
For more on "William Shakespeare: The Dramatist," and words and phrases invented by Shakespeare, please follow the links below.