Can I talk on 2 meters from the east coast to the west coast without using repeaters?
There is a list of distance records on the ARRL website. Looking at these, I would say that it might be technically possible to hold the absolute minimum of what could be called a ‘contact’ (e.g. callsign and signal report exchange) across such a distance perhaps once every ten or twenty years when exceptional conditions are all in alignment.
To answer the second part of the question, all very long distance contacts on VHF are made using highly directional antennas, or an array of such antennas (such as a multi-element Yagi, and usually take place in CW (Morse code) or possibly SSB (single sideband, voice).
Sometimes a VHF signal can be heard clearly, thousands and thousands of miles away! The usual mechanism is tropospheric ducting. It's an essentially random, temporary occurrence, not something you can rely on.
As a reliable means of direct communication, you can not expect to talk from east coast to west coast on the 2m band without using repeaters. VHF is not practical for use at distances like that, which would be in the range of 2000–3000 miles beyond the horizon.
VHF can propagate beyond its basic "radio horizon", primarily due to scattering. However, if you look at the charts in that link, you'll see that at a distance of only 500 miles there is already about 240 dB of path loss you'd need to overcome.
At the legal limit of 1500W, you could transmit a 60 dBm signal. Let's say you need a -130 dBm signal for successful receipt. (This would be an "S3 signal". For VHF, S9 is defined as -93 dBm and each S-unit is 6dB.) Since 60 dBm [the transmitted power] minus 240 dBm [the path loss] is -180 dBm, you need 50 dBm of additional gain to meet that -130 dBm goal.
To put this in perspective, consider this 10 meter long, 14 element Yagi — it's advertised to have 16.63 dBi of gain. Put one of those at each end and you're still almost 18 dB short. You might be able to add a pre-amp at the receiver, but only if the signal is still above the noise at that location. (Since we were aiming for only an S3 signal, I'm not sure how good the chances of that would be!)
I suppose you might use two parabolic dishes each in the realm of 14–16 meters in diameter to get 50 dBm of gain but remember we started with the situation at only 500 miles. The United States is about 7 dB wider than that — probably even more so as far as path loss goes. So now we're talking dishes in excess of 20m diameter, at minimum.
In short, trying to use reliable atmospheric scattering of 2m signals across the United States is probably harder than bouncing your signal off the moon, i.e. sending your signal out a distance of 1.2 light-seconds and then receiving, after it has travelled that distance back again, whatever manages to reflect off the lunar surface! (Earth-Moon-Earth communication has between 251 and 253 dB of path loss.)
(Unless, as another commenter points out, your antennas are reeeeeeeeally high. Plugging two 500,000 ft high antennas into KD4SAI's VHF/UHF Line of Sight Calculator gives you a nice 2000 mile line-of-site distance. The Friis path loss is only about 146 dBm at that range. So, raising the stations to about 95 miles high at each end does give you a much more auspicious start, link-budget-wise — if you spend enough on the mast, you could use fairly cheap antennas ;-)
My understanding is that it has been done before (and is maybe done regularly) on 2 meters via EME while operating within FCC limitations.
While this answer is not about crossing the USA/continent, there has been long distance attempts cross-Atlantic on the 2 meter band, the following will give you some info on what was attempted:
Note: the Brendan award has never been awarded, so it will be pretty amazing if a group actually manages to do a 2-way QSO crossing the Atlantic, within the rules of the award.
What is amazing, that this crowd actually managed a one-way signal at their last attempt, 3840 KM. Unfortunately this was not qualified under the Brendan rules, as you can read in the epilogue. Basically the signal was bounced off the ISS passing at the exact same moment of the (one-way) signal. As the ISS is a man-made object in space, this was excluded from the application for a Brendan Plate (the "plate" would have been awarded for the one-way contact if not for the ISS bounce.)
I hope this will give you some information in regards long-distance QSO's on the 2-meter band. It is all about amps, antenna, and mode, and a good bit of luck.
For all practical purposes,
It would take a gigantic antenna array, a lot of power, and very exceptional conditions indeed! As Scott nicely explained, "Once every ten or twenty years" is exactly right, and is almost the equivalent of a NO answer.
It would almost never happen on FM, and definitely not from a mobile!
This 2 meter antenna array was built by Dave Olean K1WHS, in Maine (owner of Directive Systems and Engineering), a long-time 2m DXer and EME op. He had that stacked and phased very-low-angle array pointed down the east coast during a 2m contest a few years ago. IIRC, even with two different kinds of exceptional propagation enhancements taking place simultaneously, and the fact that some of the farthest stations that he worked was because some of the signals were over water, he never even made it all the way down the east coast*. And most of the 2m stations he was working had towers, high-gain arrays, and linear amplifiers as well!
*(Having said that, with that nice array Dave heard so many stations that it sounded like a contest on 20 meters! :-)
Now you see what kind of challenge you face if you want to work from coast-to-coast on 2 meters.
Aside from the very rare cross-continent band openings mentioned elsewhere in answers here, you can do precisely what you say using intermediary services such as IRLP (the Internet Radio Linking Project), Allstar, or Echolink (these need not necessarily be used via repeaters), but the Internet is being used as a transport between the two ends. Even HF communications aren't consistently reliable coast to coast in North America; the continent is just too wide once you get north of the Gulf of Mexico.