All the varied answers are the result of everyone putting in their 2 cents worth of advice, some of whom missed the point of the OPs question...
Of course you can shoot without a tripod. I’m sure even an 8x10 could be hand held. The question is fundamental, though: is there a relationship between the need for a tripod to maintain sharpness and increasing pixel density (or, perhaps more correctly, decreasing pixel size).
The answer is YES. With smaller pixels (higher map or pixel density or more pixels per unit area), use of a tripod is more helpful for maintaining sharpness. Is it necessary? Depends on your shooting style, your expectations, how much you pixel peep and print size.
I agree with Terry - it all depends on one's expectations....
Me, I shoot with a high MP Sony (for landscapes anyway) AND I print big (A2+ minimum) and so IQ is of the utmost concern and priority - and so tripods, remote releases, heavy beanbags (draped over the camera), base ISO, and shutter speed (when possible) all get a workout to try and muscle IQ as much as is possible.
However, a lot of folks just want a nice low-res image to post on facebook or whatever - images that I bin would be just fine for this purpose, it just isn't my purpose...
Nonetheless, as Terry also points out, taking a slapdash approach to shooting with a high-megapixel camera is likely to disappoint - one simply cannot generally shoot with a 40 MP camera in the same way as one could with a 6 MP or 10 MP camera.
A Sony A7R series camera is smaller and lighter than any of the 6 MP or 10 MP cameras that I own from past days and by virtues of these facts lends itself to handholding - just not if IQ is the prime consideration...
Ultimately, a high megapixel camera demands careful attention to detail if one wants consistently high IQ - there is an envelope here which cannot be breached!
And, I think it is great - technique and craft still rule!!!
Tony Jay